Theorem finixpnum 37592

Description: A finite Cartesian product of numerable sets is numerable. (Contributed by Brendan Leahy, 24-Feb-2019.)

Assertion

Ref	Expression
finixpnum	⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card) → X𝑥 ∈ 𝐴 𝐵 ∈ dom card)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem finixpnum

Dummy variables 𝑣 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3321	. . . 4 ⊢ (𝑤 = ∅ → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ dom card))
2		ixpeq1 8947	. . . . . 6 ⊢ (𝑤 = ∅ → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ ∅ 𝐵)
3		ixp0x 8965	. . . . . 6 ⊢ X𝑥 ∈ ∅ 𝐵 = {∅}
4	2, 3	eqtrdi 2791	. . . . 5 ⊢ (𝑤 = ∅ → X𝑥 ∈ 𝑤 𝐵 = {∅})
5	4	eleq1d 2824	. . . 4 ⊢ (𝑤 = ∅ → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ {∅} ∈ dom card))
6	1, 5	imbi12d 344	. . 3 ⊢ (𝑤 = ∅ → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)))
7		raleq 3321	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ 𝑦 𝐵 ∈ dom card))
8		ixpeq1 8947	. . . . 5 ⊢ (𝑤 = 𝑦 → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ 𝑦 𝐵)
9	8	eleq1d 2824	. . . 4 ⊢ (𝑤 = 𝑦 → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ 𝑦 𝐵 ∈ dom card))
10	7, 9	imbi12d 344	. . 3 ⊢ (𝑤 = 𝑦 → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card)))
11		raleq 3321	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
12		ralunb 4207	. . . . . 6 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card))
13		vex 3482	. . . . . . . 8 ⊢ 𝑧 ∈ V
14		ralsnsg 4675	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ [𝑧 / 𝑥]𝐵 ∈ dom card))
15		sbcel1g 4422	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥]𝐵 ∈ dom card ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
16	14, 15	bitrd 279	. . . . . . . 8 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
17	13, 16	ax-mp 5	. . . . . . 7 ⊢ (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card)
18	17	anbi2i 623	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card) ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
19	12, 18	bitri 275	. . . . 5 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
20	11, 19	bitrdi 287	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card)))
21		ixpeq1 8947	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
22	21	eleq1d 2824	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
23	20, 22	imbi12d 344	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
24		raleq 3321	. . . 4 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card))
25		ixpeq1 8947	. . . . 5 ⊢ (𝑤 = 𝐴 → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ 𝐴 𝐵)
26	25	eleq1d 2824	. . . 4 ⊢ (𝑤 = 𝐴 → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ 𝐴 𝐵 ∈ dom card))
27	24, 26	imbi12d 344	. . 3 ⊢ (𝑤 = 𝐴 → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ dom card → X𝑥 ∈ 𝐴 𝐵 ∈ dom card)))
28		snfi 9082	. . . 4 ⊢ {∅} ∈ Fin
29		finnum 9986	. . . 4 ⊢ ({∅} ∈ Fin → {∅} ∈ dom card)
30	28, 29	mp1i 13	. . 3 ⊢ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)
31		pm2.27 42	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ 𝑦 𝐵 ∈ dom card))
32		xpnum 9989	. . . . . . . . . . 11 ⊢ ((X𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∈ dom card)
33	32	ancoms 458	. . . . . . . . . 10 ⊢ ((⦋𝑧 / 𝑥⦌𝐵 ∈ dom card ∧ X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∈ dom card)
34		xp1st 8045	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵)
35		ixpfn 8942	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵 → (1st ‘𝑤) Fn 𝑦)
36	34, 35	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (1st ‘𝑤) Fn 𝑦)
37		fvex 6920	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘𝑤) ∈ V
38	13, 37	fnsn 6626	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧}
39	36, 38	jctir 520	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → ((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧}))
40		disjsn 4716	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
41	40	biimpri 228	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∩ {𝑧}) = ∅)
42		fnun 6683	. . . . . . . . . . . . . 14 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
43	39, 41, 42	syl2anr 597	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
44		fvex 6920	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘𝑤) ∈ V
45	44	elixp 8943	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵 ↔ ((1st ‘𝑤) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
46	34, 45	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → ((1st ‘𝑤) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
47		fvun1 7000	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥 ∈ 𝑦)) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = ((1st ‘𝑤)‘𝑥))
48	38, 47	mp3an2 1448	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥 ∈ 𝑦)) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = ((1st ‘𝑤)‘𝑥))
49	48	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = ((1st ‘𝑤)‘𝑥))
50	49	eleq1d 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → ((((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
51	50	biimprd 248	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → (((1st ‘𝑤)‘𝑥) ∈ 𝐵 → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
52	51	ralimdva 3165	. . . . . . . . . . . . . . . . 17 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) → (∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
53	52	ancoms 458	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∩ {𝑧}) = ∅ ∧ (1st ‘𝑤) Fn 𝑦) → (∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
54	53	impr 454	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∩ {𝑧}) = ∅ ∧ ((1st ‘𝑤) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵)) → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
55	41, 46, 54	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
56		vsnid 4668	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ {𝑧}
57	41, 56	jctir 520	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧}))
58		fvun2 7001	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧))
59	38, 58	mp3an2 1448	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧))
60	36, 57, 59	syl2anr 597	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧))
61		csbfv 6957	. . . . . . . . . . . . . . . . 17 ⊢ ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧)
62	13, 37	fvsn 7201	. . . . . . . . . . . . . . . . . 18 ⊢ ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧) = (2nd ‘𝑤)
63	62	eqcomi 2744	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘𝑤) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧)
64	60, 61, 63	3eqtr4g 2800	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = (2nd ‘𝑤))
65		xp2nd 8046	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (2nd ‘𝑤) ∈ ⦋𝑧 / 𝑥⦌𝐵)
66	65	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → (2nd ‘𝑤) ∈ ⦋𝑧 / 𝑥⦌𝐵)
67	64, 66	eqeltrd 2839	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ ⦋𝑧 / 𝑥⦌𝐵)
68		ralsnsg 4675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ [𝑧 / 𝑥](((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
69	13, 68	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ [𝑧 / 𝑥](((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
70		sbcel12 4417	. . . . . . . . . . . . . . . 16 ⊢ ([𝑧 / 𝑥](((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ ⦋𝑧 / 𝑥⦌𝐵)
71	69, 70	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ ⦋𝑧 / 𝑥⦌𝐵)
72	67, 71	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
73		ralun 4208	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
74	55, 72, 73	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
75		snex 5442	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, (2nd ‘𝑤)⟩} ∈ V
76	44, 75	unex 7763	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) ∈ V
77	76	elixp 8943	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
78	43, 74, 77	sylanbrc 583	. . . . . . . . . . . 12 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
79	78	fmpttd 7135	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)⟶X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
80		ixpfn 8942	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → 𝑢 Fn (𝑦 ∪ {𝑧}))
81		ssun1 4188	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
82		fnssres 6692	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑢 ↾ 𝑦) Fn 𝑦)
83	80, 81, 82	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢 ↾ 𝑦) Fn 𝑦)
84		vex 3482	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑢 ∈ V
85	84	elixp 8943	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵))
86		ssralv 4064	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 (𝑢‘𝑥) ∈ 𝐵))
87	81, 86	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 (𝑢‘𝑥) ∈ 𝐵)
88		fvres 6926	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑦 → ((𝑢 ↾ 𝑦)‘𝑥) = (𝑢‘𝑥))
89	88	eleq1d 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝑦 → (((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵 ↔ (𝑢‘𝑥) ∈ 𝐵))
90	89	biimprd 248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝑦 → ((𝑢‘𝑥) ∈ 𝐵 → ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵))
91	90	ralimia 3078	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝑦 (𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
92	87, 91	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
93	92	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
94	85, 93	sylbi 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
95	84	resex 6049	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ↾ 𝑦) ∈ V
96	95	elixp 8943	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ↾ 𝑦) ∈ X𝑥 ∈ 𝑦 𝐵 ↔ ((𝑢 ↾ 𝑦) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵))
97	83, 94, 96	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢 ↾ 𝑦) ∈ X𝑥 ∈ 𝑦 𝐵)
98		ssun2 4189	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
99	98, 56	sselii 3992	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
100		csbeq1 3911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
101	100	fvixp 8941	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵 ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
102	99, 101	mpan2 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵 → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
103		nfcv 2903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤𝐵
104		nfcsb1v 3933	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
105		csbeq1a 3922	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
106	103, 104, 105	cbvixp 8953	. . . . . . . . . . . . . . . 16 ⊢ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵
107	102, 106	eleq2s 2857	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
108		opelxpi 5726	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ↾ 𝑦) ∈ X𝑥 ∈ 𝑦 𝐵 ∧ (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵) → ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵))
109	97, 107, 108	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵))
110	109	adantl 481	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵))
111		disj3 4460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ 𝑦 = (𝑦 ∖ {𝑧}))
112	40, 111	sylbb1 237	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ 𝑦 → 𝑦 = (𝑦 ∖ {𝑧}))
113		difun2 4487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
114	112, 113	eqtr4di 2793	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑧 ∈ 𝑦 → 𝑦 = ((𝑦 ∪ {𝑧}) ∖ {𝑧}))
115	114	reseq2d 6000	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑢 ↾ 𝑦) = (𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})))
116	115	uneq1d 4177	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
117	116	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
118		fvex 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢‘𝑧) ∈ V
119	95, 118	op1std 8023	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → (1st ‘𝑤) = (𝑢 ↾ 𝑦))
120	95, 118	op2ndd 8024	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → (2nd ‘𝑤) = (𝑢‘𝑧))
121	120	opeq2d 4885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → ⟨𝑧, (2nd ‘𝑤)⟩ = ⟨𝑧, (𝑢‘𝑧)⟩)
122	121	sneqd 4643	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → {⟨𝑧, (2nd ‘𝑤)⟩} = {⟨𝑧, (𝑢‘𝑧)⟩})
123	119, 122	uneq12d 4179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
124		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})) = (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))
125		snex 5442	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨𝑧, (𝑢‘𝑧)⟩} ∈ V
126	95, 125	unex 7763	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) ∈ V
127	123, 124, 126	fvmpt 7016	. . . . . . . . . . . . . . . 16 ⊢ (⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
128	109, 127	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
129	128	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
130		fnsnsplit 7204	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
131	80, 99, 130	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
132	131	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
133	117, 129, 132	3eqtr4rd 2786	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩))
134		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣) = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩))
135	134	rspceeqv 3645	. . . . . . . . . . . . 13 ⊢ ((⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩)) → ∃𝑣 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣))
136	110, 133, 135	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ∃𝑣 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣))
137	136	ralrimiva 3144	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 → ∀𝑢 ∈ X 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵∃𝑣 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣))
138		dffo3 7122	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)–onto→X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)⟶X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∧ ∀𝑢 ∈ X 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵∃𝑣 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)𝑢 = ((𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}))‘𝑣)))
139	79, 137, 138	sylanbrc 583	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)–onto→X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
140		fonum 10096	. . . . . . . . . 10 ⊢ (((X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∈ dom card ∧ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ↦ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})):(X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)–onto→X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
141	33, 139, 140	syl2anr 597	. . . . . . . . 9 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ (⦋𝑧 / 𝑥⦌𝐵 ∈ dom card ∧ X𝑥 ∈ 𝑦 𝐵 ∈ dom card)) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
142	141	expr 456	. . . . . . . 8 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
143	31, 142	syl9r 78	. . . . . . 7 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
144	143	expimpd 453	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑦 → ((⦋𝑧 / 𝑥⦌𝐵 ∈ dom card ∧ ∀𝑥 ∈ 𝑦 𝐵 ∈ dom card) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
145	144	ancomsd 465	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝑦 → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
146	145	com23 86	. . . 4 ⊢ (¬ 𝑧 ∈ 𝑦 → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
147	146	adantl 481	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
148	6, 10, 23, 27, 30, 147	findcard2s 9204	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 𝐵 ∈ dom card → X𝑥 ∈ 𝐴 𝐵 ∈ dom card))
149	148	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card) → X𝑥 ∈ 𝐴 𝐵 ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478  [wsbc 3791  ⦋csb 3908   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  {csn 4631  ⟨cop 4637   ↦ cmpt 5231   × cxp 5687  dom cdm 5689   ↾ cres 5691   Fn wfn 6558  ⟶wf 6559  –onto→wfo 6561  ‘cfv 6563  1^st c1st 8011  2^nd c2nd 8012  Xcixp 8936  Fincfn 8984  cardccrd 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-card 9977  df-acn 9980

This theorem is referenced by:  poimirlem32  37639