Theorem finixpnum 38508

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 3317	. . . 4 ⊢ (𝑤 = ∅ → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ dom card))
2		ixpeq1 8929	. . . . . 6 ⊢ (𝑤 = ∅ → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ ∅ 𝐵)
3		ixp0x 8947	. . . . . 6 ⊢ X𝑥 ∈ ∅ 𝐵 = {∅}
4	2, 3	eqtrdi 2812	. . . . 5 ⊢ (𝑤 = ∅ → X𝑥 ∈ 𝑤 𝐵 = {∅})
5	4	eleq1d 2846	. . . 4 ⊢ (𝑤 = ∅ → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ {∅} ∈ dom card))
6	1, 5	imbi12d 347	. . 3 ⊢ (𝑤 = ∅ → ((∀𝑥 ∈ 𝑤 𝐵 ∈ dom card → X𝑥 ∈ 𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)))
7		raleq 3317	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ 𝑦 𝐵 ∈ dom card))
8		ixpeq1 8929	. . . . 5 ⊢ (𝑤 = 𝑦 → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ 𝑦 𝐵)
9	8	eleq1d 2846	. . . 4 ⊢ (𝑤 = 𝑦 → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ 𝑦 𝐵 ∈ dom card))
10	7, 9	imbi12d 347	. . 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 3317	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
12		ralunb 4143	. . . . . 6 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card))
13		vex 3455	. . . . . . . 8 ⊢ 𝑧 ∈ V
14		ralsnsg 4631	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ [𝑧 / 𝑥]𝐵 ∈ dom card))
15		sbcel1g 4374	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥]𝐵 ∈ dom card ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
16	14, 15	bitrd 282	. . . . . . . 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 635	. . . . . 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 278	. . . . 5 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card))
20	11, 19	bitrdi 290	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card)))
21		ixpeq1 8929	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
22	21	eleq1d 2846	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
23	20, 22	imbi12d 347	. . 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 3317	. . . 4 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card))
25		ixpeq1 8929	. . . . 5 ⊢ (𝑤 = 𝐴 → X𝑥 ∈ 𝑤 𝐵 = X𝑥 ∈ 𝐴 𝐵)
26	25	eleq1d 2846	. . . 4 ⊢ (𝑤 = 𝐴 → (X𝑥 ∈ 𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ 𝐴 𝐵 ∈ dom card))
27	24, 26	imbi12d 347	. . 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 9064	. . . 4 ⊢ {∅} ∈ Fin
29		finnum 10022	. . . 4 ⊢ ({∅} ∈ Fin → {∅} ∈ dom card)
30	28, 29	mp1i 14	. . 3 ⊢ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)
31		pm2.27 43	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → ((∀𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → X𝑥 ∈ 𝑦 𝐵 ∈ dom card))
32		xpnum 10025	. . . . . . . . . . 11 ⊢ ((X𝑥 ∈ 𝑦 𝐵 ∈ dom card ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∈ dom card)
33	32	ancoms 464	. . . . . . . . . 10 ⊢ ((⦋𝑧 / 𝑥⦌𝐵 ∈ dom card ∧ X𝑥 ∈ 𝑦 𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) ∈ dom card)
34		xp1st 8031	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵)
35		ixpfn 8924	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵 → (1st ‘𝑤) Fn 𝑦)
36	34, 35	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (1st ‘𝑤) Fn 𝑦)
37		fvex 6896	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘𝑤) ∈ V
38	13, 37	fnsn 6596	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧}
39	36, 38	jctir 530	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → ((1st ‘𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd ‘𝑤)⟩} Fn {𝑧}))
40		disjsn 4672	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
41	40	biimpri 231	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑦 ∩ {𝑧}) = ∅)
42		fnun 6651	. . . . . . . . . . . . . 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 609	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
44		fvex 6896	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘𝑤) ∈ V
45	44	elixp 8925	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑤) ∈ X𝑥 ∈ 𝑦 𝐵 ↔ ((1st ‘𝑤) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
46	34, 45	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → ((1st ‘𝑤) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
47		fvun1 6974	. . . . . . . . . . . . . . . . . . . . . 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 1478	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥 ∈ 𝑦)) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = ((1st ‘𝑤)‘𝑥))
49	48	anassrs 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = ((1st ‘𝑤)‘𝑥))
50	49	eleq1d 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → ((((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ((1st ‘𝑤)‘𝑥) ∈ 𝐵))
51	50	biimprd 251	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1st ‘𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥 ∈ 𝑦) → (((1st ‘𝑤)‘𝑥) ∈ 𝐵 → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵))
52	51	ralimdva 3175	. . . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ 𝑦 (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
56		vsnid 4624	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ {𝑧}
57	41, 56	jctir 530	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧}))
58		fvun2 6975	. . . . . . . . . . . . . . . . . . 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 1478	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧))
60	36, 57, 59	syl2anr 609	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧))
61		csbfv 6930	. . . . . . . . . . . . . . . . 17 ⊢ ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑧)
62	13, 37	fvsn 7184	. . . . . . . . . . . . . . . . . 18 ⊢ ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧) = (2nd ‘𝑤)
63	62	eqcomi 2770	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘𝑤) = ({⟨𝑧, (2nd ‘𝑤)⟩}‘𝑧)
64	60, 61, 63	3eqtr4g 2821	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) = (2nd ‘𝑤))
65		xp2nd 8032	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵) → (2nd ‘𝑤) ∈ ⦋𝑧 / 𝑥⦌𝐵)
66	65	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → (2nd ‘𝑤) ∈ ⦋𝑧 / 𝑥⦌𝐵)
67	64, 66	eqeltrd 2861	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ⦋𝑧 / 𝑥⦌(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ ⦋𝑧 / 𝑥⦌𝐵)
68		ralsnsg 4631	. . . . . . . . . . . . . . . . 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 4369	. . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ {𝑧} (((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
73		ralun 4144	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩})‘𝑥) ∈ 𝐵)
75		snex 5397	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, (2nd ‘𝑤)⟩} ∈ V
76	44, 75	unex 7759	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) ∈ V
77	76	elixp 8925	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 12 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑤 ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵)) → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
79	78	fmpttd 7113	. . . . . . . . . . 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 8924	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → 𝑢 Fn (𝑦 ∪ {𝑧}))
81		ssun1 4124	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
82		fnssres 6660	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑢 ↾ 𝑦) Fn 𝑦)
83	80, 81, 82	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢 ↾ 𝑦) Fn 𝑦)
84		vex 3455	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑢 ∈ V
85	84	elixp 8925	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵))
86		ssralv 4000	. . . . . . . . . . . . . . . . . . . 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 6902	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑦 → ((𝑢 ↾ 𝑦)‘𝑥) = (𝑢‘𝑥))
89	88	eleq1d 2846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝑦 → (((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵 ↔ (𝑢‘𝑥) ∈ 𝐵))
90	89	biimprd 251	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝑦 → ((𝑢‘𝑥) ∈ 𝐵 → ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵))
91	90	ralimia 3097	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝑦 (𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
92	87, 91	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
93	92	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
94	85, 93	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵)
95	84	resex 6018	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ↾ 𝑦) ∈ V
96	95	elixp 8925	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ↾ 𝑦) ∈ X𝑥 ∈ 𝑦 𝐵 ↔ ((𝑢 ↾ 𝑦) Fn 𝑦 ∧ ∀𝑥 ∈ 𝑦 ((𝑢 ↾ 𝑦)‘𝑥) ∈ 𝐵))
97	83, 94, 96	sylanbrc 595	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢 ↾ 𝑦) ∈ X𝑥 ∈ 𝑦 𝐵)
98		ssun2 4125	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
99	98, 56	sselii 3928	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
100		csbeq1 3850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
101	100	fvixp 8923	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵 ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
102	99, 101	mpan2 704	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵 → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
103		nfcv 2923	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤𝐵
104		nfcsb1v 3871	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
105		csbeq1a 3861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
106	103, 104, 105	cbvixp 8935	. . . . . . . . . . . . . . . 16 ⊢ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = X𝑤 ∈ (𝑦 ∪ {𝑧})⦋𝑤 / 𝑥⦌𝐵
107	102, 106	eleq2s 2879	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢‘𝑧) ∈ ⦋𝑧 / 𝑥⦌𝐵)
108		opelxpi 5688	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵))
110	109	adantl 487	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ ∈ (X𝑥 ∈ 𝑦 𝐵 × ⦋𝑧 / 𝑥⦌𝐵))
111		disj3 4407	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ 𝑦 = (𝑦 ∖ {𝑧}))
112	40, 111	sylbb1 240	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ 𝑦 → 𝑦 = (𝑦 ∖ {𝑧}))
113		difun2 4437	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
114	112, 113	eqtr4di 2814	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑧 ∈ 𝑦 → 𝑦 = ((𝑦 ∪ {𝑧}) ∖ {𝑧}))
115	114	reseq2d 5970	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑢 ↾ 𝑦) = (𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})))
116	115	uneq1d 4114	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
117	116	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
118		fvex 6896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢‘𝑧) ∈ V
119	95, 118	op1std 8009	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → (1st ‘𝑤) = (𝑢 ↾ 𝑦))
120	95, 118	op2ndd 8010	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → (2nd ‘𝑤) = (𝑢‘𝑧))
121	120	opeq2d 4840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → ⟨𝑧, (2nd ‘𝑤)⟩ = ⟨𝑧, (𝑢‘𝑧)⟩)
122	121	sneqd 4596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → {⟨𝑧, (2nd ‘𝑤)⟩} = {⟨𝑧, (𝑢‘𝑧)⟩})
123	119, 122	uneq12d 4116	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ⟨(𝑢 ↾ 𝑦), (𝑢‘𝑧)⟩ → ((1st ‘𝑤) ∪ {⟨𝑧, (2nd ‘𝑤)⟩}) = ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
124		eqid 2761	. . . . . . . . . . . . . . . . 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 5397	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨𝑧, (𝑢‘𝑧)⟩} ∈ V
126	95, 125	unex 7759	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ↾ 𝑦) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}) ∈ V
127	123, 124, 126	fvmpt 6991	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 7187	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
131	80, 99, 130	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
132	131	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ 𝑢 ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢‘𝑧)⟩}))
133	117, 129, 132	3eqtr4rd 2807	. . . . . . . . . . . . 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 6883	. . . . . . . . . . . . . 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 3599	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 3155	. . . . . . . . . . 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 7100	. . . . . . . . . . 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 595	. . . . . . . . . 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 10130	. . . . . . . . . 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 609	. . . . . . . . 9 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ (⦋𝑧 / 𝑥⦌𝐵 ∈ dom card ∧ X𝑥 ∈ 𝑦 𝐵 ∈ dom card)) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
142	141	expr 462	. . . . . . . 8 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ dom card) → (X𝑥 ∈ 𝑦 𝐵 ∈ dom card → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
143	31, 142	syl9r 79	. . . . . . 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 459	. . . . . 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 471	. . . . 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 87	. . . 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 487	. . 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 9174	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 𝐵 ∈ dom card → X𝑥 ∈ 𝐴 𝐵 ∈ dom card))
149	148	imp 412	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ dom card) → X𝑥 ∈ 𝐴 𝐵 ∈ dom card)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451  [wsbc 3739  ⦋csb 3847   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584  ⟨cop 4590   ↦ cmpt 5186   × cxp 5649  dom cdm 5651   ↾ cres 5653   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  1^st c1st 7997  2^nd c2nd 7998  Xcixp 8918  Fincfn 8966  cardccrd 10009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-omul 8474  df-er 8710  df-map 8842  df-ixp 8919  df-en 8967  df-dom 8968  df-fin 8970  df-card 10013  df-acn 10016

This theorem is used by:  poimirlem32  38550