Theorem xpassen 6058

Description: Associative law for equinumerosity of cross product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by SF, 24-Feb-2015.)

Hypotheses

Ref	Expression
xpassen.1	⊢ A ∈ V
xpassen.2	⊢ B ∈ V
xpassen.3	⊢ C ∈ V

Assertion

Ref	Expression
xpassen	⊢ ((A × B) × C) ≈ (A × (B × C))

Proof of Theorem xpassen

Dummy variables a b c x y z t p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1stfo 5506	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
2		fof 5270	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → 1st :V–→V)
3	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ 1st :V–→V
4		dffn2 5225	. . . . . . . . . 10 ⊢ (1st Fn V ↔ 1st :V–→V)
5	3, 4	mpbir 200	. . . . . . . . 9 ⊢ 1st Fn V
6		ssv 3292	. . . . . . . . 9 ⊢ ran 1st ⊆ V
7		fnco 5192	. . . . . . . . 9 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
8	5, 5, 6, 7	mp3an 1277	. . . . . . . 8 ⊢ (1st ∘ 1st ) Fn V
9		2ndfo 5507	. . . . . . . . . . . 12 ⊢ 2nd :V–onto→V
10		fofn 5272	. . . . . . . . . . . 12 ⊢ (2nd :V–onto→V → 2nd Fn V)
11	9, 10	ax-mp 5	. . . . . . . . . . 11 ⊢ 2nd Fn V
12		fnco 5192	. . . . . . . . . . 11 ⊢ ((2nd Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (2nd ∘ 1st ) Fn V)
13	11, 5, 6, 12	mp3an 1277	. . . . . . . . . 10 ⊢ (2nd ∘ 1st ) Fn V
14		fntxp 5805	. . . . . . . . . 10 ⊢ (((2nd ∘ 1st ) Fn V ∧ 2nd Fn V) → ((2nd ∘ 1st ) ⊗ 2nd ) Fn (V ∩ V))
15	13, 11, 14	mp2an 653	. . . . . . . . 9 ⊢ ((2nd ∘ 1st ) ⊗ 2nd ) Fn (V ∩ V)
16		inidm 3465	. . . . . . . . . 10 ⊢ (V ∩ V) = V
17	16	fneq2i 5180	. . . . . . . . 9 ⊢ (((2nd ∘ 1st ) ⊗ 2nd ) Fn (V ∩ V) ↔ ((2nd ∘ 1st ) ⊗ 2nd ) Fn V)
18	15, 17	mpbi 199	. . . . . . . 8 ⊢ ((2nd ∘ 1st ) ⊗ 2nd ) Fn V
19		fntxp 5805	. . . . . . . 8 ⊢ (((1st ∘ 1st ) Fn V ∧ ((2nd ∘ 1st ) ⊗ 2nd ) Fn V) → ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) Fn (V ∩ V))
20	8, 18, 19	mp2an 653	. . . . . . 7 ⊢ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) Fn (V ∩ V)
21	16	fneq2i 5180	. . . . . . 7 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) Fn (V ∩ V) ↔ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) Fn V)
22	20, 21	mpbi 199	. . . . . 6 ⊢ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) Fn V
23		dffn2 5225	. . . . . 6 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) Fn V ↔ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )):V–→V)
24	22, 23	mpbi 199	. . . . 5 ⊢ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )):V–→V
25		xpassenlem 6057	. . . . . . . . 9 ⊢ (y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ↔ ( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x))
26		xpassenlem 6057	. . . . . . . . 9 ⊢ (z((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ↔ ( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x))
27		simp1 955	. . . . . . . . . . . . . 14 ⊢ (( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) → Proj1 Proj1 y = Proj1 x)
28		simp1 955	. . . . . . . . . . . . . 14 ⊢ (( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x) → Proj1 Proj1 z = Proj1 x)
29		eqtr3 2372	. . . . . . . . . . . . . 14 ⊢ (( Proj1 Proj1 y = Proj1 x ∧ Proj1 Proj1 z = Proj1 x) → Proj1 Proj1 y = Proj1 Proj1 z)
30	27, 28, 29	syl2an 463	. . . . . . . . . . . . 13 ⊢ ((( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) ∧ ( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x)) → Proj1 Proj1 y = Proj1 Proj1 z)
31		simp2 956	. . . . . . . . . . . . . 14 ⊢ (( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) → Proj2 Proj1 y = Proj1 Proj2 x)
32		simp2 956	. . . . . . . . . . . . . 14 ⊢ (( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x) → Proj2 Proj1 z = Proj1 Proj2 x)
33		eqtr3 2372	. . . . . . . . . . . . . 14 ⊢ (( Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 Proj1 z = Proj1 Proj2 x) → Proj2 Proj1 y = Proj2 Proj1 z)
34	31, 32, 33	syl2an 463	. . . . . . . . . . . . 13 ⊢ ((( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) ∧ ( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x)) → Proj2 Proj1 y = Proj2 Proj1 z)
35	30, 34	opeq12d 4587	. . . . . . . . . . . 12 ⊢ ((( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) ∧ ( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x)) → ⟨ Proj1 Proj1 y, Proj2 Proj1 y⟩ = ⟨ Proj1 Proj1 z, Proj2 Proj1 z⟩)
36		opeq 4620	. . . . . . . . . . . 12 ⊢ Proj1 y = ⟨ Proj1 Proj1 y, Proj2 Proj1 y⟩
37		opeq 4620	. . . . . . . . . . . 12 ⊢ Proj1 z = ⟨ Proj1 Proj1 z, Proj2 Proj1 z⟩
38	35, 36, 37	3eqtr4g 2410	. . . . . . . . . . 11 ⊢ ((( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) ∧ ( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x)) → Proj1 y = Proj1 z)
39		simp3 957	. . . . . . . . . . . 12 ⊢ (( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) → Proj2 y = Proj2 Proj2 x)
40		simp3 957	. . . . . . . . . . . 12 ⊢ (( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x) → Proj2 z = Proj2 Proj2 x)
41		eqtr3 2372	. . . . . . . . . . . 12 ⊢ (( Proj2 y = Proj2 Proj2 x ∧ Proj2 z = Proj2 Proj2 x) → Proj2 y = Proj2 z)
42	39, 40, 41	syl2an 463	. . . . . . . . . . 11 ⊢ ((( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) ∧ ( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x)) → Proj2 y = Proj2 z)
43	38, 42	opeq12d 4587	. . . . . . . . . 10 ⊢ ((( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) ∧ ( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x)) → ⟨ Proj1 y, Proj2 y⟩ = ⟨ Proj1 z, Proj2 z⟩)
44		opeq 4620	. . . . . . . . . 10 ⊢ y = ⟨ Proj1 y, Proj2 y⟩
45		opeq 4620	. . . . . . . . . 10 ⊢ z = ⟨ Proj1 z, Proj2 z⟩
46	43, 44, 45	3eqtr4g 2410	. . . . . . . . 9 ⊢ ((( Proj1 Proj1 y = Proj1 x ∧ Proj2 Proj1 y = Proj1 Proj2 x ∧ Proj2 y = Proj2 Proj2 x) ∧ ( Proj1 Proj1 z = Proj1 x ∧ Proj2 Proj1 z = Proj1 Proj2 x ∧ Proj2 z = Proj2 Proj2 x)) → y = z)
47	25, 26, 46	syl2anb 465	. . . . . . . 8 ⊢ ((y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ∧ z((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x) → y = z)
48	47	gen2 1547	. . . . . . 7 ⊢ ∀y∀z((y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ∧ z((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x) → y = z)
49		breq1 4643	. . . . . . . 8 ⊢ (y = z → (y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ↔ z((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x))
50	49	mo4 2237	. . . . . . 7 ⊢ (∃*y y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ↔ ∀y∀z((y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x ∧ z((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x) → y = z))
51	48, 50	mpbir 200	. . . . . 6 ⊢ ∃*y y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x
52	51	ax-gen 1546	. . . . 5 ⊢ ∀x∃*y y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x
53		dff12 5258	. . . . 5 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )):V–1-1→V ↔ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )):V–→V ∧ ∀x∃*y y((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))x))
54	24, 52, 53	mpbir2an 886	. . . 4 ⊢ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )):V–1-1→V
55		ssv 3292	. . . 4 ⊢ ((A × B) × C) ⊆ V
56		f1ores 5301	. . . 4 ⊢ ((((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )):V–1-1→V ∧ ((A × B) × C) ⊆ V) → (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)):((A × B) × C)–1-1-onto→(((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)))
57	54, 55, 56	mp2an 653	. . 3 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)):((A × B) × C)–1-1-onto→(((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C))
58		vex 2863	. . . . . 6 ⊢ p ∈ V
59		opeqex 4622	. . . . . 6 ⊢ (p ∈ V → ∃b∃c p = ⟨b, c⟩)
60		rexcom4 2879	. . . . . . . . . . . 12 ⊢ (∃x ∈ A ∃p∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃p∃x ∈ A ∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
61		rexcom4 2879	. . . . . . . . . . . . . 14 ⊢ (∃y ∈ B ∃p∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃p∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
62		rexcom4 2879	. . . . . . . . . . . . . . . 16 ⊢ (∃z ∈ C ∃p(p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃p∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
63		vex 2863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ x ∈ V
64		vex 2863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ y ∈ V
65	63, 64	opex 4589	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨x, y⟩ ∈ V
66		vex 2863	. . . . . . . . . . . . . . . . . . . 20 ⊢ z ∈ V
67	65, 66	opex 4589	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨⟨x, y⟩, z⟩ ∈ V
68		breq1 4643	. . . . . . . . . . . . . . . . . . 19 ⊢ (p = ⟨⟨x, y⟩, z⟩ → (p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩ ↔ ⟨⟨x, y⟩, z⟩((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
69	67, 68	ceqsexv 2895	. . . . . . . . . . . . . . . . . 18 ⊢ (∃p(p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ⟨⟨x, y⟩, z⟩((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩)
70	65, 66	brco1st 5778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨⟨x, y⟩, z⟩(1st ∘ 1st )a ↔ ⟨x, y⟩1st a)
71	63, 64	opbr1st 5502	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨x, y⟩1st a ↔ x = a)
72	70, 71	bitri 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨⟨x, y⟩, z⟩(1st ∘ 1st )a ↔ x = a)
73		trtxp 5782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨⟨x, y⟩, z⟩((2nd ∘ 1st ) ⊗ 2nd )⟨b, c⟩ ↔ (⟨⟨x, y⟩, z⟩(2nd ∘ 1st )b ∧ ⟨⟨x, y⟩, z⟩2nd c))
74		brco 4884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨⟨x, y⟩, z⟩(2nd ∘ 1st )b ↔ ∃t(⟨⟨x, y⟩, z⟩1st t ∧ t2nd b))
75	65, 66	opbr1st 5502	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨⟨x, y⟩, z⟩1st t ↔ ⟨x, y⟩ = t)
76		eqcom 2355	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨x, y⟩ = t ↔ t = ⟨x, y⟩)
77	75, 76	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨⟨x, y⟩, z⟩1st t ↔ t = ⟨x, y⟩)
78	77	anbi1i 676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⟨⟨x, y⟩, z⟩1st t ∧ t2nd b) ↔ (t = ⟨x, y⟩ ∧ t2nd b))
79	78	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃t(⟨⟨x, y⟩, z⟩1st t ∧ t2nd b) ↔ ∃t(t = ⟨x, y⟩ ∧ t2nd b))
80		breq1 4643	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (t = ⟨x, y⟩ → (t2nd b ↔ ⟨x, y⟩2nd b))
81	65, 80	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃t(t = ⟨x, y⟩ ∧ t2nd b) ↔ ⟨x, y⟩2nd b)
82	63, 64	opbr2nd 5503	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨x, y⟩2nd b ↔ y = b)
83	81, 82	bitri 240	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃t(t = ⟨x, y⟩ ∧ t2nd b) ↔ y = b)
84	74, 79, 83	3bitri 262	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨⟨x, y⟩, z⟩(2nd ∘ 1st )b ↔ y = b)
85	65, 66	opbr2nd 5503	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨⟨x, y⟩, z⟩2nd c ↔ z = c)
86	84, 85	anbi12i 678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨⟨x, y⟩, z⟩(2nd ∘ 1st )b ∧ ⟨⟨x, y⟩, z⟩2nd c) ↔ (y = b ∧ z = c))
87	73, 86	bitri 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨⟨x, y⟩, z⟩((2nd ∘ 1st ) ⊗ 2nd )⟨b, c⟩ ↔ (y = b ∧ z = c))
88	72, 87	anbi12i 678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨⟨x, y⟩, z⟩(1st ∘ 1st )a ∧ ⟨⟨x, y⟩, z⟩((2nd ∘ 1st ) ⊗ 2nd )⟨b, c⟩) ↔ (x = a ∧ (y = b ∧ z = c)))
89		trtxp 5782	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨⟨x, y⟩, z⟩((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩ ↔ (⟨⟨x, y⟩, z⟩(1st ∘ 1st )a ∧ ⟨⟨x, y⟩, z⟩((2nd ∘ 1st ) ⊗ 2nd )⟨b, c⟩))
90		3anass 938	. . . . . . . . . . . . . . . . . . 19 ⊢ ((x = a ∧ y = b ∧ z = c) ↔ (x = a ∧ (y = b ∧ z = c)))
91	88, 89, 90	3bitr4i 268	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨⟨x, y⟩, z⟩((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩ ↔ (x = a ∧ y = b ∧ z = c))
92	69, 91	bitri 240	. . . . . . . . . . . . . . . . 17 ⊢ (∃p(p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ (x = a ∧ y = b ∧ z = c))
93	92	rexbii 2640	. . . . . . . . . . . . . . . 16 ⊢ (∃z ∈ C ∃p(p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃z ∈ C (x = a ∧ y = b ∧ z = c))
94	62, 93	bitr3i 242	. . . . . . . . . . . . . . 15 ⊢ (∃p∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃z ∈ C (x = a ∧ y = b ∧ z = c))
95	94	rexbii 2640	. . . . . . . . . . . . . 14 ⊢ (∃y ∈ B ∃p∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃y ∈ B ∃z ∈ C (x = a ∧ y = b ∧ z = c))
96	61, 95	bitr3i 242	. . . . . . . . . . . . 13 ⊢ (∃p∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃y ∈ B ∃z ∈ C (x = a ∧ y = b ∧ z = c))
97	96	rexbii 2640	. . . . . . . . . . . 12 ⊢ (∃x ∈ A ∃p∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃x ∈ A ∃y ∈ B ∃z ∈ C (x = a ∧ y = b ∧ z = c))
98	60, 97	bitr3i 242	. . . . . . . . . . 11 ⊢ (∃p∃x ∈ A ∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃x ∈ A ∃y ∈ B ∃z ∈ C (x = a ∧ y = b ∧ z = c))
99		3reeanv 2780	. . . . . . . . . . 11 ⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C (x = a ∧ y = b ∧ z = c) ↔ (∃x ∈ A x = a ∧ ∃y ∈ B y = b ∧ ∃z ∈ C z = c))
100	98, 99	bitri 240	. . . . . . . . . 10 ⊢ (∃p∃x ∈ A ∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ (∃x ∈ A x = a ∧ ∃y ∈ B y = b ∧ ∃z ∈ C z = c))
101		r19.41v 2765	. . . . . . . . . . . 12 ⊢ (∃x ∈ A (∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ (∃x ∈ A ∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
102		r19.41v 2765	. . . . . . . . . . . . . . 15 ⊢ (∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ (∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
103	102	rexbii 2640	. . . . . . . . . . . . . 14 ⊢ (∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃y ∈ B (∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
104		r19.41v 2765	. . . . . . . . . . . . . 14 ⊢ (∃y ∈ B (∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ (∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
105	103, 104	bitri 240	. . . . . . . . . . . . 13 ⊢ (∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ (∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
106	105	rexbii 2640	. . . . . . . . . . . 12 ⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃x ∈ A (∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
107		elxp2 4803	. . . . . . . . . . . . . 14 ⊢ (p ∈ ((A × B) × C) ↔ ∃t ∈ (A × B)∃z ∈ C p = ⟨t, z⟩)
108		df-rex 2621	. . . . . . . . . . . . . 14 ⊢ (∃t ∈ (A × B)∃z ∈ C p = ⟨t, z⟩ ↔ ∃t(t ∈ (A × B) ∧ ∃z ∈ C p = ⟨t, z⟩))
109		rexcom4 2879	. . . . . . . . . . . . . . 15 ⊢ (∃x ∈ A ∃t∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩) ↔ ∃t∃x ∈ A ∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
110		opeq1 4579	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (t = ⟨x, y⟩ → ⟨t, z⟩ = ⟨⟨x, y⟩, z⟩)
111	110	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (t = ⟨x, y⟩ → (p = ⟨t, z⟩ ↔ p = ⟨⟨x, y⟩, z⟩))
112	65, 111	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃t(t = ⟨x, y⟩ ∧ p = ⟨t, z⟩) ↔ p = ⟨⟨x, y⟩, z⟩)
113	112	rexbii 2640	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃z ∈ C ∃t(t = ⟨x, y⟩ ∧ p = ⟨t, z⟩) ↔ ∃z ∈ C p = ⟨⟨x, y⟩, z⟩)
114		rexcom4 2879	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃z ∈ C ∃t(t = ⟨x, y⟩ ∧ p = ⟨t, z⟩) ↔ ∃t∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
115	113, 114	bitr3i 242	. . . . . . . . . . . . . . . . . 18 ⊢ (∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ↔ ∃t∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
116	115	rexbii 2640	. . . . . . . . . . . . . . . . 17 ⊢ (∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ↔ ∃y ∈ B ∃t∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
117		rexcom4 2879	. . . . . . . . . . . . . . . . 17 ⊢ (∃y ∈ B ∃t∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩) ↔ ∃t∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
118	116, 117	bitri 240	. . . . . . . . . . . . . . . 16 ⊢ (∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ↔ ∃t∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
119	118	rexbii 2640	. . . . . . . . . . . . . . 15 ⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ↔ ∃x ∈ A ∃t∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
120		r19.41v 2765	. . . . . . . . . . . . . . . . 17 ⊢ (∃x ∈ A (∃y ∈ B t = ⟨x, y⟩ ∧ ∃z ∈ C p = ⟨t, z⟩) ↔ (∃x ∈ A ∃y ∈ B t = ⟨x, y⟩ ∧ ∃z ∈ C p = ⟨t, z⟩))
121		reeanv 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩) ↔ (∃y ∈ B t = ⟨x, y⟩ ∧ ∃z ∈ C p = ⟨t, z⟩))
122	121	rexbii 2640	. . . . . . . . . . . . . . . . 17 ⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩) ↔ ∃x ∈ A (∃y ∈ B t = ⟨x, y⟩ ∧ ∃z ∈ C p = ⟨t, z⟩))
123		elxp2 4803	. . . . . . . . . . . . . . . . . 18 ⊢ (t ∈ (A × B) ↔ ∃x ∈ A ∃y ∈ B t = ⟨x, y⟩)
124	123	anbi1i 676	. . . . . . . . . . . . . . . . 17 ⊢ ((t ∈ (A × B) ∧ ∃z ∈ C p = ⟨t, z⟩) ↔ (∃x ∈ A ∃y ∈ B t = ⟨x, y⟩ ∧ ∃z ∈ C p = ⟨t, z⟩))
125	120, 122, 124	3bitr4ri 269	. . . . . . . . . . . . . . . 16 ⊢ ((t ∈ (A × B) ∧ ∃z ∈ C p = ⟨t, z⟩) ↔ ∃x ∈ A ∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
126	125	exbii 1582	. . . . . . . . . . . . . . 15 ⊢ (∃t(t ∈ (A × B) ∧ ∃z ∈ C p = ⟨t, z⟩) ↔ ∃t∃x ∈ A ∃y ∈ B ∃z ∈ C (t = ⟨x, y⟩ ∧ p = ⟨t, z⟩))
127	109, 119, 126	3bitr4ri 269	. . . . . . . . . . . . . 14 ⊢ (∃t(t ∈ (A × B) ∧ ∃z ∈ C p = ⟨t, z⟩) ↔ ∃x ∈ A ∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩)
128	107, 108, 127	3bitri 262	. . . . . . . . . . . . 13 ⊢ (p ∈ ((A × B) × C) ↔ ∃x ∈ A ∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩)
129	128	anbi1i 676	. . . . . . . . . . . 12 ⊢ ((p ∈ ((A × B) × C) ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ (∃x ∈ A ∃y ∈ B ∃z ∈ C p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
130	101, 106, 129	3bitr4ri 269	. . . . . . . . . . 11 ⊢ ((p ∈ ((A × B) × C) ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃x ∈ A ∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
131	130	exbii 1582	. . . . . . . . . 10 ⊢ (∃p(p ∈ ((A × B) × C) ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ ∃p∃x ∈ A ∃y ∈ B ∃z ∈ C (p = ⟨⟨x, y⟩, z⟩ ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
132		risset 2662	. . . . . . . . . . 11 ⊢ (a ∈ A ↔ ∃x ∈ A x = a)
133		risset 2662	. . . . . . . . . . 11 ⊢ (b ∈ B ↔ ∃y ∈ B y = b)
134		risset 2662	. . . . . . . . . . 11 ⊢ (c ∈ C ↔ ∃z ∈ C z = c)
135	132, 133, 134	3anbi123i 1140	. . . . . . . . . 10 ⊢ ((a ∈ A ∧ b ∈ B ∧ c ∈ C) ↔ (∃x ∈ A x = a ∧ ∃y ∈ B y = b ∧ ∃z ∈ C z = c))
136	100, 131, 135	3bitr4i 268	. . . . . . . . 9 ⊢ (∃p(p ∈ ((A × B) × C) ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩) ↔ (a ∈ A ∧ b ∈ B ∧ c ∈ C))
137		elima2 4756	. . . . . . . . 9 ⊢ (⟨a, ⟨b, c⟩⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ ∃p(p ∈ ((A × B) × C) ∧ p((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd ))⟨a, ⟨b, c⟩⟩))
138		opelxp 4812	. . . . . . . . . . 11 ⊢ (⟨b, c⟩ ∈ (B × C) ↔ (b ∈ B ∧ c ∈ C))
139	138	anbi2i 675	. . . . . . . . . 10 ⊢ ((a ∈ A ∧ ⟨b, c⟩ ∈ (B × C)) ↔ (a ∈ A ∧ (b ∈ B ∧ c ∈ C)))
140		opelxp 4812	. . . . . . . . . 10 ⊢ (⟨a, ⟨b, c⟩⟩ ∈ (A × (B × C)) ↔ (a ∈ A ∧ ⟨b, c⟩ ∈ (B × C)))
141		3anass 938	. . . . . . . . . 10 ⊢ ((a ∈ A ∧ b ∈ B ∧ c ∈ C) ↔ (a ∈ A ∧ (b ∈ B ∧ c ∈ C)))
142	139, 140, 141	3bitr4i 268	. . . . . . . . 9 ⊢ (⟨a, ⟨b, c⟩⟩ ∈ (A × (B × C)) ↔ (a ∈ A ∧ b ∈ B ∧ c ∈ C))
143	136, 137, 142	3bitr4i 268	. . . . . . . 8 ⊢ (⟨a, ⟨b, c⟩⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ ⟨a, ⟨b, c⟩⟩ ∈ (A × (B × C)))
144		opeq2 4580	. . . . . . . . . 10 ⊢ (p = ⟨b, c⟩ → ⟨a, p⟩ = ⟨a, ⟨b, c⟩⟩)
145	144	eleq1d 2419	. . . . . . . . 9 ⊢ (p = ⟨b, c⟩ → (⟨a, p⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ ⟨a, ⟨b, c⟩⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C))))
146	144	eleq1d 2419	. . . . . . . . 9 ⊢ (p = ⟨b, c⟩ → (⟨a, p⟩ ∈ (A × (B × C)) ↔ ⟨a, ⟨b, c⟩⟩ ∈ (A × (B × C))))
147	145, 146	bibi12d 312	. . . . . . . 8 ⊢ (p = ⟨b, c⟩ → ((⟨a, p⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ ⟨a, p⟩ ∈ (A × (B × C))) ↔ (⟨a, ⟨b, c⟩⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ ⟨a, ⟨b, c⟩⟩ ∈ (A × (B × C)))))
148	143, 147	mpbiri 224	. . . . . . 7 ⊢ (p = ⟨b, c⟩ → (⟨a, p⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ ⟨a, p⟩ ∈ (A × (B × C))))
149	148	exlimivv 1635	. . . . . 6 ⊢ (∃b∃c p = ⟨b, c⟩ → (⟨a, p⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ ⟨a, p⟩ ∈ (A × (B × C))))
150	58, 59, 149	mp2b 9	. . . . 5 ⊢ (⟨a, p⟩ ∈ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ ⟨a, p⟩ ∈ (A × (B × C)))
151	150	eqrelriv 4851	. . . 4 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) = (A × (B × C))
152		f1oeq3 5284	. . . 4 ⊢ ((((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) = (A × (B × C)) → ((((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)):((A × B) × C)–1-1-onto→(((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)):((A × B) × C)–1-1-onto→(A × (B × C))))
153	151, 152	ax-mp 5	. . 3 ⊢ ((((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)):((A × B) × C)–1-1-onto→(((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) “ ((A × B) × C)) ↔ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)):((A × B) × C)–1-1-onto→(A × (B × C)))
154	57, 153	mpbi 199	. 2 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)):((A × B) × C)–1-1-onto→(A × (B × C))
155		1stex 4740	. . . . . 6 ⊢ 1st ∈ V
156	155, 155	coex 4751	. . . . 5 ⊢ (1st ∘ 1st ) ∈ V
157		2ndex 5113	. . . . . . 7 ⊢ 2nd ∈ V
158	157, 155	coex 4751	. . . . . 6 ⊢ (2nd ∘ 1st ) ∈ V
159	158, 157	txpex 5786	. . . . 5 ⊢ ((2nd ∘ 1st ) ⊗ 2nd ) ∈ V
160	156, 159	txpex 5786	. . . 4 ⊢ ((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ∈ V
161		xpassen.1	. . . . . 6 ⊢ A ∈ V
162		xpassen.2	. . . . . 6 ⊢ B ∈ V
163	161, 162	xpex 5116	. . . . 5 ⊢ (A × B) ∈ V
164		xpassen.3	. . . . 5 ⊢ C ∈ V
165	163, 164	xpex 5116	. . . 4 ⊢ ((A × B) × C) ∈ V
166	160, 165	resex 5118	. . 3 ⊢ (((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)) ∈ V
167	166	f1oen 6034	. 2 ⊢ ((((1st ∘ 1st ) ⊗ ((2nd ∘ 1st ) ⊗ 2nd )) ↾ ((A × B) × C)):((A × B) × C)–1-1-onto→(A × (B × C)) → ((A × B) × C) ≈ (A × (B × C)))
168	154, 167	ax-mp 5	1 ⊢ ((A × B) × C) ≈ (A × (B × C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∃wrex 2616  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  ⟨cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722   “ cima 4723   × cxp 4771  ran crn 4774   ↾ cres 4775   Fn wfn 4777  –→wf 4778  –1-1→wf1 4779  –onto→wfo 4780  –1-1-onto→wf1o 4781  2^nd c2nd 4784   ⊗ ctxp 5736   ≈ cen 6029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-en 6030

This theorem is referenced by:  mucass  6136