Theorem fmucndlem 23443

Description: Lemma for fmucnd 23444. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Assertion

Ref	Expression
fmucndlem	⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦

Proof of Theorem fmucndlem

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 5602	. . 3 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ran ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴))
2		simpr 485	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
3		resmpo 7394	. . . . 5 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩))
4	2, 3	sylancom 588	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩))
5	4	rneqd 5847	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ran ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↾ (𝐴 × 𝐴)) = ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩))
6	1, 5	eqtrid 2790	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩))
7		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
9	7, 8	op1std 7841	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑝) = 𝑥)
10	9	fveq2d 6778	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑝)) = (𝐹‘𝑥))
11	7, 8	op2ndd 7842	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑝) = 𝑦)
12	11	fveq2d 6778	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑝)) = (𝐹‘𝑦))
13	10, 12	opeq12d 4812	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
14	13	mpompt 7388	. . . . . . . . 9 ⊢ (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
15	14	eqcomi 2747	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
16	15	rneqi 5846	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = ran (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
17		fvexd 6789	. . . . . . 7 ⊢ ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(1st ‘𝑝)) ∈ V)
18		fvexd 6789	. . . . . . 7 ⊢ ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(2nd ‘𝑝)) ∈ V)
19	16, 17, 18	fliftrel 7179	. . . . . 6 ⊢ (⊤ → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⊆ (V × V))
20	19	mptru 1546	. . . . 5 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ⊆ (V × V)
21	20	sseli 3917	. . . 4 ⊢ (𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) → 𝑝 ∈ (V × V))
22	21	adantl 482	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)) → 𝑝 ∈ (V × V))
23		xpss 5605	. . . . 5 ⊢ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ⊆ (V × V)
24	23	sseli 3917	. . . 4 ⊢ (𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) → 𝑝 ∈ (V × V))
25	24	adantl 482	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))) → 𝑝 ∈ (V × V))
26		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
27		opex 5379	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
28	26, 27	elrnmpo 7410	. . . . . . . 8 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
29		eqcom 2745	. . . . . . . . . 10 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
30		fvex 6787	. . . . . . . . . . 11 ⊢ (1st ‘𝑝) ∈ V
31		fvex 6787	. . . . . . . . . . 11 ⊢ (2nd ‘𝑝) ∈ V
32	30, 31	opth2 5395	. . . . . . . . . 10 ⊢ (⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ↔ ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
33	29, 32	bitri 274	. . . . . . . . 9 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
34	33	2rexbii 3182	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)))
35		reeanv 3294	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (1st ‘𝑝) ∧ (𝐹‘𝑦) = (2nd ‘𝑝)) ↔ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝) ∧ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝)))
36	28, 34, 35	3bitri 297	. . . . . . 7 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝) ∧ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝)))
37		fvelimab 6841	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝)))
38		fvelimab 6841	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((2nd ‘𝑝) ∈ (𝐹 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝)))
39	37, 38	anbi12d 631	. . . . . . 7 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → (((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ∧ (2nd ‘𝑝) ∈ (𝐹 “ 𝐴)) ↔ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = (1st ‘𝑝) ∧ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = (2nd ‘𝑝))))
40	36, 39	bitr4id 290	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ∧ (2nd ‘𝑝) ∈ (𝐹 “ 𝐴))))
41		opelxp 5625	. . . . . 6 ⊢ (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ↔ ((1st ‘𝑝) ∈ (𝐹 “ 𝐴) ∧ (2nd ‘𝑝) ∈ (𝐹 “ 𝐴)))
42	40, 41	bitr4di 289	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))))
43	42	adantr 481	. . . 4 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → (⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))))
44		1st2nd2 7870	. . . . . 6 ⊢ (𝑝 ∈ (V × V) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
45	44	adantl 482	. . . . 5 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
46	45	eleq1d 2823	. . . 4 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)))
47	45	eleq1d 2823	. . . 4 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)) ↔ ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))))
48	43, 46, 47	3bitr4d 311	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ↔ 𝑝 ∈ ((𝐹 “ 𝐴) × (𝐹 “ 𝐴))))
49	22, 25, 48	eqrdav 2737	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))
50	6, 49	eqtrd 2778	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ⟨cop 4567   ↦ cmpt 5157   × cxp 5587  ran crn 5590   ↾ cres 5591   “ cima 5592   Fn wfn 6428  ‘cfv 6433   ∈ cmpo 7277  1^st c1st 7829  2^nd c2nd 7830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832

This theorem is referenced by:  fmucnd  23444