Theorem oemapvali 9637

Description: If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
oemapval.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
oemapval.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
oemapvali.r	⊢ (𝜑 → 𝐹𝑇𝐺)
oemapvali.x	⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}

Assertion

Ref	Expression
oemapvali	⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))

Distinct variable groups:   𝑤,𝑐,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐   𝑤,𝐹,𝑥,𝑦,𝑧   𝑆,𝑐,𝑥,𝑦,𝑧   𝐺,𝑐,𝑤,𝑥,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑤,𝑋,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐

Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑋(𝑐)

Proof of Theorem oemapvali

Step	Hyp	Ref	Expression
1		oemapvali.r	. . 3 ⊢ (𝜑 → 𝐹𝑇𝐺)
2		cantnfs.s	. . . 4 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
3		cantnfs.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ On)
4		cantnfs.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ On)
5		oemapval.t	. . . 4 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
6		oemapval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
7		oemapval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
8	2, 3, 4, 5, 6, 7	oemapval 9636	. . 3 ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
9	1, 8	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
10		ssrab2 4043	. . . 4 ⊢ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ 𝐵
11		oemapvali.x	. . . . 5 ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}
12	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐵 ∈ On)
13		onss 7761	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
14	12, 13	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐵 ⊆ On)
15	10, 14	sstrid 3958	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ On)
16	2, 3, 4	cantnfs 9619	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
17	7, 16	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
18	17	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝐺 finSupp ∅)
19	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐺 finSupp ∅)
20	4	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝐵 ∈ On)
21		simp2 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝑐 ∈ 𝐵)
22	17	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
23	22	ffnd 6689	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Fn 𝐵)
24	23	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝐺 Fn 𝐵)
25		ne0i 4304	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑐) ∈ (𝐺‘𝑐) → (𝐺‘𝑐) ≠ ∅)
26	25	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → (𝐺‘𝑐) ≠ ∅)
27		fvn0elsupp 8159	. . . . . . . . . 10 ⊢ (((𝐵 ∈ On ∧ 𝑐 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑐) ≠ ∅)) → 𝑐 ∈ (𝐺 supp ∅))
28	20, 21, 24, 26, 27	syl22anc 838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ (𝐹‘𝑐) ∈ (𝐺‘𝑐)) → 𝑐 ∈ (𝐺 supp ∅))
29	28	rabssdv 4038	. . . . . . . 8 ⊢ (𝜑 → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅))
30	29	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅))
31		fsuppimp 9319	. . . . . . . 8 ⊢ (𝐺 finSupp ∅ → (Fun 𝐺 ∧ (𝐺 supp ∅) ∈ Fin))
32		ssfi 9137	. . . . . . . . 9 ⊢ (((𝐺 supp ∅) ∈ Fin ∧ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅)) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin)
33	32	ex 412	. . . . . . . 8 ⊢ ((𝐺 supp ∅) ∈ Fin → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin))
34	31, 33	simpl2im 503	. . . . . . 7 ⊢ (𝐺 finSupp ∅ → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ (𝐺 supp ∅) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin))
35	19, 30, 34	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin)
36		fveq2 6858	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝐹‘𝑐) = (𝐹‘𝑧))
37		fveq2 6858	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (𝐺‘𝑐) = (𝐺‘𝑧))
38	36, 37	eleq12d 2822	. . . . . . . 8 ⊢ (𝑐 = 𝑧 → ((𝐹‘𝑐) ∈ (𝐺‘𝑐) ↔ (𝐹‘𝑧) ∈ (𝐺‘𝑧)))
39		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ 𝐵)
40		simprrl 780	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐹‘𝑧) ∈ (𝐺‘𝑧))
41	38, 39, 40	elrabd 3661	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
42	41	ne0d 4305	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ≠ ∅)
43		ordunifi 9237	. . . . . 6 ⊢ (({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ On ∧ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ Fin ∧ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ≠ ∅) → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
44	15, 35, 42, 43	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
45	11, 44	eqeltrid 2832	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
46	10, 45	sselid 3944	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ 𝐵)
47		fveq2 6858	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
48		fveq2 6858	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
49	47, 48	eleq12d 2822	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ∈ (𝐺‘𝑥) ↔ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
50		fveq2 6858	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (𝐹‘𝑐) = (𝐹‘𝑥))
51		fveq2 6858	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (𝐺‘𝑐) = (𝐺‘𝑥))
52	50, 51	eleq12d 2822	. . . . . . 7 ⊢ (𝑐 = 𝑥 → ((𝐹‘𝑐) ∈ (𝐺‘𝑐) ↔ (𝐹‘𝑥) ∈ (𝐺‘𝑥)))
53	52	cbvrabv 3416	. . . . . 6 ⊢ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} = {𝑥 ∈ 𝐵 ∣ (𝐹‘𝑥) ∈ (𝐺‘𝑥)}
54	49, 53	elrab2 3662	. . . . 5 ⊢ (𝑋 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ↔ (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
55	45, 54	sylib 218	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋)))
56	55	simprd 495	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐹‘𝑋) ∈ (𝐺‘𝑋))
57		simprrr 781	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
58	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐴 ∈ On)
59	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝐺:𝐵⟶𝐴)
60	59, 46	ffvelcdmd 7057	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐺‘𝑋) ∈ 𝐴)
61		onelon 6357	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑋) ∈ 𝐴) → (𝐺‘𝑋) ∈ On)
62	58, 60, 61	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝐺‘𝑋) ∈ On)
63		eloni 6342	. . . . . . . . . 10 ⊢ ((𝐺‘𝑋) ∈ On → Ord (𝐺‘𝑋))
64		ordirr 6350	. . . . . . . . . 10 ⊢ (Ord (𝐺‘𝑋) → ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋))
65	62, 63, 64	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋))
66		nelneq 2852	. . . . . . . . 9 ⊢ (((𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ¬ (𝐺‘𝑋) ∈ (𝐺‘𝑋)) → ¬ (𝐹‘𝑋) = (𝐺‘𝑋))
67	56, 65, 66	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ (𝐹‘𝑋) = (𝐺‘𝑋))
68		eleq2 2817	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑋))
69		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝐹‘𝑤) = (𝐹‘𝑋))
70		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝐺‘𝑤) = (𝐺‘𝑋))
71	69, 70	eqeq12d 2745	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → ((𝐹‘𝑤) = (𝐺‘𝑤) ↔ (𝐹‘𝑋) = (𝐺‘𝑋)))
72	68, 71	imbi12d 344	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → ((𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ (𝑧 ∈ 𝑋 → (𝐹‘𝑋) = (𝐺‘𝑋))))
73	72, 57, 46	rspcdva 3589	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑧 ∈ 𝑋 → (𝐹‘𝑋) = (𝐺‘𝑋)))
74	67, 73	mtod 198	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ¬ 𝑧 ∈ 𝑋)
75		ssexg 5278	. . . . . . . . . . 11 ⊢ (({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ 𝐵 ∧ 𝐵 ∈ On) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V)
76	10, 12, 75	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V)
77		ssonuni 7756	. . . . . . . . . 10 ⊢ ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ V → ({𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⊆ On → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ On))
78	76, 15, 77	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ∈ On)
79	11, 78	eqeltrid 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ∈ On)
80		onelon 6357	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ On)
81	12, 39, 80	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ∈ On)
82		ontri1 6366	. . . . . . . 8 ⊢ ((𝑋 ∈ On ∧ 𝑧 ∈ On) → (𝑋 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑋))
83	79, 81, 82	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑋 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑋))
84	74, 83	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 ⊆ 𝑧)
85		elssuni 4901	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} → 𝑧 ⊆ ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)})
86	85, 11	sseqtrrdi 3988	. . . . . . 7 ⊢ (𝑧 ∈ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} → 𝑧 ⊆ 𝑋)
87	41, 86	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑧 ⊆ 𝑋)
88	84, 87	eqssd 3964	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → 𝑋 = 𝑧)
89		eleq1 2816	. . . . . . 7 ⊢ (𝑋 = 𝑧 → (𝑋 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
90	89	imbi1d 341	. . . . . 6 ⊢ (𝑋 = 𝑧 → ((𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
91	90	ralbidv 3156	. . . . 5 ⊢ (𝑋 = 𝑧 → (∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
92	88, 91	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
93	57, 92	mpbird 257	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))
94	46, 56, 93	3jca 1128	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
95	9, 94	rexlimddv 3140	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  ∪ cuni 4871   class class class wbr 5107  {copab 5169  dom cdm 5638  Ord word 6331  Oncon0 6332  Fun wfun 6505   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   supp csupp 8139  Fincfn 8918   finSupp cfsupp 9312   CNF ccnf 9614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-seqom 8416  df-1o 8434  df-map 8801  df-en 8919  df-fin 8922  df-fsupp 9313  df-cnf 9615

This theorem is referenced by:  cantnflem1a  9638  cantnflem1b  9639  cantnflem1c  9640  cantnflem1d  9641  cantnflem1  9642