Theorem ovcelem1 6172

Description: Lemma for ovce 6173. Set up stratification for the result. (Contributed by SF, 6-Mar-2015.)

Assertion

Ref	Expression
ovcelem1	⊢ ((N ∈ V ∧ M ∈ W) → {g ∣ ∃a∃b(℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b))} ∈ V)

Distinct variable groups:   a,b,g,M   N,a,b,g

Allowed substitution hints:   V(g, a, b)   W(g, a, b)

Proof of Theorem ovcelem1

Dummy variables f t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elima1c 4948	. . . 4 ⊢ (g ∈ ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) “ 1c) ↔ ∃a⟨{a}, g⟩ ∈ (( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c))
2		elima1c 4948	. . . . . 6 ⊢ (⟨{a}, g⟩ ∈ (( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) ↔ ∃b⟨{b}, ⟨{a}, g⟩⟩ ∈ ( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )))
3		vex 2863	. . . . . . . . . . 11 ⊢ g ∈ V
4	3	otelins3 5793	. . . . . . . . . 10 ⊢ (⟨{b}, ⟨{a}, g⟩⟩ ∈ Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ↔ ⟨{b}, {a}⟩ ∈ ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)))
5		opelcnv 4894	. . . . . . . . . 10 ⊢ (⟨{b}, {a}⟩ ∈ ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ↔ ⟨{a}, {b}⟩ ∈ ((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)))
6		opelxp 4812	. . . . . . . . . . 11 ⊢ (⟨{a}, {b}⟩ ∈ ((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ↔ ({a} ∈ (◡ Pw1Fn “ N) ∧ {b} ∈ (◡ Pw1Fn “ M)))
7		brcnv 4893	. . . . . . . . . . . . . . 15 ⊢ (t◡ Pw1Fn {a} ↔ {a} Pw1Fn t)
8		vex 2863	. . . . . . . . . . . . . . . 16 ⊢ a ∈ V
9	8	brpw1fn 5855	. . . . . . . . . . . . . . 15 ⊢ ({a} Pw1Fn t ↔ t = ℘1a)
10	7, 9	bitri 240	. . . . . . . . . . . . . 14 ⊢ (t◡ Pw1Fn {a} ↔ t = ℘1a)
11	10	rexbii 2640	. . . . . . . . . . . . 13 ⊢ (∃t ∈ N t◡ Pw1Fn {a} ↔ ∃t ∈ N t = ℘1a)
12		elima 4755	. . . . . . . . . . . . 13 ⊢ ({a} ∈ (◡ Pw1Fn “ N) ↔ ∃t ∈ N t◡ Pw1Fn {a})
13		risset 2662	. . . . . . . . . . . . 13 ⊢ (℘1a ∈ N ↔ ∃t ∈ N t = ℘1a)
14	11, 12, 13	3bitr4i 268	. . . . . . . . . . . 12 ⊢ ({a} ∈ (◡ Pw1Fn “ N) ↔ ℘1a ∈ N)
15		brcnv 4893	. . . . . . . . . . . . . . 15 ⊢ (t◡ Pw1Fn {b} ↔ {b} Pw1Fn t)
16		vex 2863	. . . . . . . . . . . . . . . 16 ⊢ b ∈ V
17	16	brpw1fn 5855	. . . . . . . . . . . . . . 15 ⊢ ({b} Pw1Fn t ↔ t = ℘1b)
18	15, 17	bitri 240	. . . . . . . . . . . . . 14 ⊢ (t◡ Pw1Fn {b} ↔ t = ℘1b)
19	18	rexbii 2640	. . . . . . . . . . . . 13 ⊢ (∃t ∈ M t◡ Pw1Fn {b} ↔ ∃t ∈ M t = ℘1b)
20		elima 4755	. . . . . . . . . . . . 13 ⊢ ({b} ∈ (◡ Pw1Fn “ M) ↔ ∃t ∈ M t◡ Pw1Fn {b})
21		risset 2662	. . . . . . . . . . . . 13 ⊢ (℘1b ∈ M ↔ ∃t ∈ M t = ℘1b)
22	19, 20, 21	3bitr4i 268	. . . . . . . . . . . 12 ⊢ ({b} ∈ (◡ Pw1Fn “ M) ↔ ℘1b ∈ M)
23	14, 22	anbi12i 678	. . . . . . . . . . 11 ⊢ (({a} ∈ (◡ Pw1Fn “ N) ∧ {b} ∈ (◡ Pw1Fn “ M)) ↔ (℘1a ∈ N ∧ ℘1b ∈ M))
24	6, 23	bitri 240	. . . . . . . . . 10 ⊢ (⟨{a}, {b}⟩ ∈ ((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ↔ (℘1a ∈ N ∧ ℘1b ∈ M))
25	4, 5, 24	3bitri 262	. . . . . . . . 9 ⊢ (⟨{b}, ⟨{a}, g⟩⟩ ∈ Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ↔ (℘1a ∈ N ∧ ℘1b ∈ M))
26		elrn2 4898	. . . . . . . . . 10 ⊢ (⟨{b}, ⟨{a}, g⟩⟩ ∈ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ) ↔ ∃t⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ))
27		elin 3220	. . . . . . . . . . . 12 ⊢ (⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ) ↔ (⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∧ ⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ Ins2 Ins2 ◡ ≈ ))
28		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ t ∈ V
29		snex 4112	. . . . . . . . . . . . . . . . . 18 ⊢ {b} ∈ V
30		snex 4112	. . . . . . . . . . . . . . . . . 18 ⊢ {a} ∈ V
31	29, 30	opex 4589	. . . . . . . . . . . . . . . . 17 ⊢ ⟨{b}, {a}⟩ ∈ V
32	28, 31	opex 4589	. . . . . . . . . . . . . . . 16 ⊢ ⟨t, ⟨{b}, {a}⟩⟩ ∈ V
33	32	elcompl 3226	. . . . . . . . . . . . . . 15 ⊢ (⟨t, ⟨{b}, {a}⟩⟩ ∈ ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ ¬ ⟨t, ⟨{b}, {a}⟩⟩ ∈ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c))
34		elima1c 4948	. . . . . . . . . . . . . . . . 17 ⊢ (⟨t, ⟨{b}, {a}⟩⟩ ∈ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ ∃f⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ ( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))))
35		elsymdif 3224	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ ( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) ↔ ¬ (⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ Ins3 S ↔ ⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))))
36	31	otelins3 5793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ Ins3 S ↔ ⟨{f}, t⟩ ∈ S )
37		vex 2863	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ f ∈ V
38	37, 28	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{f}, t⟩ ∈ S ↔ f ∈ t)
39	36, 38	bitri 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ Ins3 S ↔ f ∈ t)
40	28	otelins2 5792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd )) ↔ ⟨{f}, ⟨{b}, {a}⟩⟩ ∈ SI3 ( Fns ⊗ ( S ∘ Image2nd )))
41	37, 16, 8	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{f}, ⟨{b}, {a}⟩⟩ ∈ SI3 ( Fns ⊗ ( S ∘ Image2nd )) ↔ ⟨f, ⟨b, a⟩⟩ ∈ ( Fns ⊗ ( S ∘ Image2nd )))
42		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (f Fns b ↔ ⟨f, b⟩ ∈ Fns )
43	37	brfns 5834	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (f Fns b ↔ f Fn b)
44	42, 43	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨f, b⟩ ∈ Fns ↔ f Fn b)
45		opelco 4885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨f, a⟩ ∈ ( S ∘ Image2nd ) ↔ ∃t(fImage2nd t ∧ t S a))
46	37, 28	brimage 5794	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (fImage2nd t ↔ t = (2nd “ f))
47		dfrn5 5509	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ran f = (2nd “ f)
48	47	eqeq2i 2363	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (t = ran f ↔ t = (2nd “ f))
49	46, 48	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (fImage2nd t ↔ t = ran f)
50	28, 8	brsset 4759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (t S a ↔ t ⊆ a)
51	49, 50	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((fImage2nd t ∧ t S a) ↔ (t = ran f ∧ t ⊆ a))
52	51	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃t(fImage2nd t ∧ t S a) ↔ ∃t(t = ran f ∧ t ⊆ a))
53	37	rnex 5108	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ran f ∈ V
54		sseq1 3293	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (t = ran f → (t ⊆ a ↔ ran f ⊆ a))
55	53, 54	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃t(t = ran f ∧ t ⊆ a) ↔ ran f ⊆ a)
56	45, 52, 55	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨f, a⟩ ∈ ( S ∘ Image2nd ) ↔ ran f ⊆ a)
57	44, 56	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((⟨f, b⟩ ∈ Fns ∧ ⟨f, a⟩ ∈ ( S ∘ Image2nd )) ↔ (f Fn b ∧ ran f ⊆ a))
58		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨f, ⟨b, a⟩⟩ ∈ ( Fns ⊗ ( S ∘ Image2nd )) ↔ (⟨f, b⟩ ∈ Fns ∧ ⟨f, a⟩ ∈ ( S ∘ Image2nd )))
59		df-f 4792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (f:b–→a ↔ (f Fn b ∧ ran f ⊆ a))
60	57, 58, 59	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨f, ⟨b, a⟩⟩ ∈ ( Fns ⊗ ( S ∘ Image2nd )) ↔ f:b–→a)
61	40, 41, 60	3bitri 262	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd )) ↔ f:b–→a)
62	39, 61	bibi12i 306	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ Ins3 S ↔ ⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) ↔ (f ∈ t ↔ f:b–→a))
63	35, 62	xchbinx 301	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ ( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) ↔ ¬ (f ∈ t ↔ f:b–→a))
64	63	exbii 1582	. . . . . . . . . . . . . . . . 17 ⊢ (∃f⟨{f}, ⟨t, ⟨{b}, {a}⟩⟩⟩ ∈ ( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) ↔ ∃f ¬ (f ∈ t ↔ f:b–→a))
65		exnal 1574	. . . . . . . . . . . . . . . . 17 ⊢ (∃f ¬ (f ∈ t ↔ f:b–→a) ↔ ¬ ∀f(f ∈ t ↔ f:b–→a))
66	34, 64, 65	3bitrri 263	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀f(f ∈ t ↔ f:b–→a) ↔ ⟨t, ⟨{b}, {a}⟩⟩ ∈ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c))
67	66	con1bii 321	. . . . . . . . . . . . . . 15 ⊢ (¬ ⟨t, ⟨{b}, {a}⟩⟩ ∈ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ ∀f(f ∈ t ↔ f:b–→a))
68	33, 67	bitri 240	. . . . . . . . . . . . . 14 ⊢ (⟨t, ⟨{b}, {a}⟩⟩ ∈ ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ ∀f(f ∈ t ↔ f:b–→a))
69	3	oqelins4 5795	. . . . . . . . . . . . . 14 ⊢ (⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ ⟨t, ⟨{b}, {a}⟩⟩ ∈ ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c))
70	8, 16	mapval 6012	. . . . . . . . . . . . . . . 16 ⊢ (a ↑m b) = {f ∣ f:b–→a}
71	70	eqeq2i 2363	. . . . . . . . . . . . . . 15 ⊢ (t = (a ↑m b) ↔ t = {f ∣ f:b–→a})
72		eqabb 2459	. . . . . . . . . . . . . . 15 ⊢ (t = {f ∣ f:b–→a} ↔ ∀f(f ∈ t ↔ f:b–→a))
73	71, 72	bitri 240	. . . . . . . . . . . . . 14 ⊢ (t = (a ↑m b) ↔ ∀f(f ∈ t ↔ f:b–→a))
74	68, 69, 73	3bitr4i 268	. . . . . . . . . . . . 13 ⊢ (⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ t = (a ↑m b))
75	29	otelins2 5792	. . . . . . . . . . . . . 14 ⊢ (⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ Ins2 Ins2 ◡ ≈ ↔ ⟨t, ⟨{a}, g⟩⟩ ∈ Ins2 ◡ ≈ )
76	30	otelins2 5792	. . . . . . . . . . . . . 14 ⊢ (⟨t, ⟨{a}, g⟩⟩ ∈ Ins2 ◡ ≈ ↔ ⟨t, g⟩ ∈ ◡ ≈ )
77		df-br 4641	. . . . . . . . . . . . . . 15 ⊢ (t◡ ≈ g ↔ ⟨t, g⟩ ∈ ◡ ≈ )
78		brcnv 4893	. . . . . . . . . . . . . . 15 ⊢ (t◡ ≈ g ↔ g ≈ t)
79	77, 78	bitr3i 242	. . . . . . . . . . . . . 14 ⊢ (⟨t, g⟩ ∈ ◡ ≈ ↔ g ≈ t)
80	75, 76, 79	3bitri 262	. . . . . . . . . . . . 13 ⊢ (⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ Ins2 Ins2 ◡ ≈ ↔ g ≈ t)
81	74, 80	anbi12i 678	. . . . . . . . . . . 12 ⊢ ((⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∧ ⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ Ins2 Ins2 ◡ ≈ ) ↔ (t = (a ↑m b) ∧ g ≈ t))
82	27, 81	bitri 240	. . . . . . . . . . 11 ⊢ (⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ) ↔ (t = (a ↑m b) ∧ g ≈ t))
83	82	exbii 1582	. . . . . . . . . 10 ⊢ (∃t⟨t, ⟨{b}, ⟨{a}, g⟩⟩⟩ ∈ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ) ↔ ∃t(t = (a ↑m b) ∧ g ≈ t))
84		ovex 5552	. . . . . . . . . . 11 ⊢ (a ↑m b) ∈ V
85		breq2 4644	. . . . . . . . . . 11 ⊢ (t = (a ↑m b) → (g ≈ t ↔ g ≈ (a ↑m b)))
86	84, 85	ceqsexv 2895	. . . . . . . . . 10 ⊢ (∃t(t = (a ↑m b) ∧ g ≈ t) ↔ g ≈ (a ↑m b))
87	26, 83, 86	3bitri 262	. . . . . . . . 9 ⊢ (⟨{b}, ⟨{a}, g⟩⟩ ∈ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ) ↔ g ≈ (a ↑m b))
88	25, 87	anbi12i 678	. . . . . . . 8 ⊢ ((⟨{b}, ⟨{a}, g⟩⟩ ∈ Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∧ ⟨{b}, ⟨{a}, g⟩⟩ ∈ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) ↔ ((℘1a ∈ N ∧ ℘1b ∈ M) ∧ g ≈ (a ↑m b)))
89		elin 3220	. . . . . . . 8 ⊢ (⟨{b}, ⟨{a}, g⟩⟩ ∈ ( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) ↔ (⟨{b}, ⟨{a}, g⟩⟩ ∈ Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∧ ⟨{b}, ⟨{a}, g⟩⟩ ∈ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )))
90		df-3an 936	. . . . . . . 8 ⊢ ((℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b)) ↔ ((℘1a ∈ N ∧ ℘1b ∈ M) ∧ g ≈ (a ↑m b)))
91	88, 89, 90	3bitr4i 268	. . . . . . 7 ⊢ (⟨{b}, ⟨{a}, g⟩⟩ ∈ ( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) ↔ (℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b)))
92	91	exbii 1582	. . . . . 6 ⊢ (∃b⟨{b}, ⟨{a}, g⟩⟩ ∈ ( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) ↔ ∃b(℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b)))
93	2, 92	bitri 240	. . . . 5 ⊢ (⟨{a}, g⟩ ∈ (( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) ↔ ∃b(℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b)))
94	93	exbii 1582	. . . 4 ⊢ (∃a⟨{a}, g⟩ ∈ (( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) ↔ ∃a∃b(℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b)))
95	1, 94	bitri 240	. . 3 ⊢ (g ∈ ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) “ 1c) ↔ ∃a∃b(℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b)))
96	95	eqabi 2465	. 2 ⊢ ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) “ 1c) = {g ∣ ∃a∃b(℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b))}
97		pw1fnex 5853	. . . . . 6 ⊢ Pw1Fn ∈ V
98	97	cnvex 5103	. . . . 5 ⊢ ◡ Pw1Fn ∈ V
99		imaexg 4747	. . . . 5 ⊢ ((◡ Pw1Fn ∈ V ∧ N ∈ V) → (◡ Pw1Fn “ N) ∈ V)
100	98, 99	mpan 651	. . . 4 ⊢ (N ∈ V → (◡ Pw1Fn “ N) ∈ V)
101		imaexg 4747	. . . . 5 ⊢ ((◡ Pw1Fn ∈ V ∧ M ∈ W) → (◡ Pw1Fn “ M) ∈ V)
102	98, 101	mpan 651	. . . 4 ⊢ (M ∈ W → (◡ Pw1Fn “ M) ∈ V)
103		xpexg 5115	. . . 4 ⊢ (((◡ Pw1Fn “ N) ∈ V ∧ (◡ Pw1Fn “ M) ∈ V) → ((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V)
104	100, 102, 103	syl2an 463	. . 3 ⊢ ((N ∈ V ∧ M ∈ W) → ((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V)
105		cnvexg 5102	. . . 4 ⊢ (((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V → ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V)
106		ins3exg 5797	. . . 4 ⊢ (◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V → Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V)
107	105, 106	syl 15	. . 3 ⊢ (((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V → Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V)
108		ssetex 4745	. . . . . . . . . . . 12 ⊢ S ∈ V
109	108	ins3ex 5799	. . . . . . . . . . 11 ⊢ Ins3 S ∈ V
110		fnsex 5833	. . . . . . . . . . . . . 14 ⊢ Fns ∈ V
111		2ndex 5113	. . . . . . . . . . . . . . . 16 ⊢ 2nd ∈ V
112	111	imageex 5802	. . . . . . . . . . . . . . 15 ⊢ Image2nd ∈ V
113	108, 112	coex 4751	. . . . . . . . . . . . . 14 ⊢ ( S ∘ Image2nd ) ∈ V
114	110, 113	txpex 5786	. . . . . . . . . . . . 13 ⊢ ( Fns ⊗ ( S ∘ Image2nd )) ∈ V
115	114	si3ex 5807	. . . . . . . . . . . 12 ⊢ SI3 ( Fns ⊗ ( S ∘ Image2nd )) ∈ V
116	115	ins2ex 5798	. . . . . . . . . . 11 ⊢ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd )) ∈ V
117	109, 116	symdifex 4109	. . . . . . . . . 10 ⊢ ( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) ∈ V
118		1cex 4143	. . . . . . . . . 10 ⊢ 1c ∈ V
119	117, 118	imaex 4748	. . . . . . . . 9 ⊢ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∈ V
120	119	complex 4105	. . . . . . . 8 ⊢ ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∈ V
121	120	ins4ex 5800	. . . . . . 7 ⊢ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∈ V
122		enex 6032	. . . . . . . . . 10 ⊢ ≈ ∈ V
123	122	cnvex 5103	. . . . . . . . 9 ⊢ ◡ ≈ ∈ V
124	123	ins2ex 5798	. . . . . . . 8 ⊢ Ins2 ◡ ≈ ∈ V
125	124	ins2ex 5798	. . . . . . 7 ⊢ Ins2 Ins2 ◡ ≈ ∈ V
126	121, 125	inex 4106	. . . . . 6 ⊢ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ) ∈ V
127	126	rnex 5108	. . . . 5 ⊢ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ) ∈ V
128		inexg 4101	. . . . 5 ⊢ (( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V ∧ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ ) ∈ V) → ( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) ∈ V)
129	127, 128	mpan2 652	. . . 4 ⊢ ( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V → ( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) ∈ V)
130		imaexg 4747	. . . . 5 ⊢ ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) ∈ V ∧ 1c ∈ V) → (( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) ∈ V)
131	118, 130	mpan2 652	. . . 4 ⊢ (( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) ∈ V → (( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) ∈ V)
132		imaexg 4747	. . . . 5 ⊢ (((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) ∈ V ∧ 1c ∈ V) → ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) “ 1c) ∈ V)
133	118, 132	mpan2 652	. . . 4 ⊢ ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) ∈ V → ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) “ 1c) ∈ V)
134	129, 131, 133	3syl 18	. . 3 ⊢ ( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∈ V → ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) “ 1c) ∈ V)
135	104, 107, 134	3syl 18	. 2 ⊢ ((N ∈ V ∧ M ∈ W) → ((( Ins3 ◡((◡ Pw1Fn “ N) × (◡ Pw1Fn “ M)) ∩ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ◡ ≈ )) “ 1c) “ 1c) ∈ V)
136	96, 135	syl5eqelr 2438	1 ⊢ ((N ∈ V ∧ M ∈ W) → {g ∣ ∃a∃b(℘1a ∈ N ∧ ℘1b ∈ M ∧ g ≈ (a ↑m b))} ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∩ cin 3209   ⊕ csymdif 3210   ⊆ wss 3258  {csn 3738  1_cc1c 4135  ℘₁cpw1 4136  ⟨cop 4562   class class class wbr 4640   S csset 4720   ∘ ccom 4722   “ cima 4723   × cxp 4771  ^◡ccnv 4772  ran crn 4774   Fn wfn 4777  –→wf 4778  2^nd c2nd 4784  (class class class)co 5526   ⊗ ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752  Imagecimage 5754   Ins4 cins4 5756   SI₃ csi3 5758   Fns cfns 5762   Pw1Fn cpw1fn 5766   ↑_m cmap 6000   ≈ cen 6029

This proof depends on 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 proof 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-sset 4726  df-co 4727  df-ima 4728  df-si 4729  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-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-map 6002  df-en 6030

This theorem is used by:  ovce  6173  fnce  6177