Theorem ac6num 10584

Description: A version of ac6 10585 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)

Hypothesis

Ref	Expression
ac6num.1	⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ac6num	⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))

Distinct variable groups:   𝑥,𝑓,𝐴   𝑦,𝑓,𝐵,𝑥   𝜑,𝑓   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑓)   𝐴(𝑦)   𝑉(𝑥, 𝑦, 𝑓)

Proof of Theorem ac6num

Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfiu1 4989	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑}
2	1	nfel1 2942	. . . . . . . 8 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card
3		ssiun2 5009	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
4		ssexg 5285	. . . . . . . . . 10 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card) → {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
5	4	expcom 419	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ({𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} → {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V))
6	3, 5	syl5 35	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V))
7	2, 6	ralrimi 3264	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
8		dfiun2g 4991	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}})
9	7, 8	syl 18	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}})
10		eqid 2763	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})
11	10	rnmpt 5946	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}}
12	11	unieqi 4882	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}}
13	9, 12	eqtr4di 2816	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}))
14		id 23	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card)
15	13, 14	eqeltrrd 2865	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ dom card)
16	15	3ad2ant2 1152	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ dom card)
17		simp3 1156	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
18		necom 3012	. . . . . . . 8 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦 ∈ 𝐵 ∣ 𝜑})
19		rabn0 4342	. . . . . . . 8 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
20		df-ne 2960	. . . . . . . 8 ⊢ (∅ ≠ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
21	18, 19, 20	3bitr3i 304	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
22	21	ralbii 3112	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
23		ralnex 3092	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
24	22, 23	bitri 278	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
25	17, 24	sylib 221	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
26		0ex 5264	. . . . 5 ⊢ ∅ ∈ V
27	10	elrnmpt 5947	. . . . 5 ⊢ (∅ ∈ V → (∅ ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑}))
28	26, 27	ax-mp 5	. . . 4 ⊢ (∅ ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
29	25, 28	sylnibr 332	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ¬ ∅ ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}))
30		ac5num 10142	. . 3 ⊢ ((∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ dom card ∧ ¬ ∅ ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})) → ∃𝑔(𝑔:ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})⟶∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧))
31	16, 29, 30	syl2anc 596	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑔(𝑔:ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})⟶∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧))
32		ffn 6731	. . . . . 6 ⊢ (𝑔:ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})⟶∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) → 𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}))
33	32	anim1i 627	. . . . 5 ⊢ ((𝑔:ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})⟶∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧) → (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧))
34	7	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
35		fveq2 6908	. . . . . . . . 9 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑔‘𝑧) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
36		id 23	. . . . . . . . 9 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑})
37	35, 36	eleq12d 2858	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ((𝑔‘𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}))
38	10, 37	ralrnmptw 7118	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}))
39	34, 38	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}))
40	39	anbi2d 642	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ((𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧) ↔ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})))
41	33, 40	imbitrid 247	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ((𝑔:ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})⟶∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧) → (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})))
42		simpl1 1210	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})) → 𝐴 ∈ 𝑉)
43	42	mptexd 7253	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∈ V)
44		elrabi 3644	. . . . . . . . . 10 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
45	44	ralimi 3103	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
46	45	ad2antll 742	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})) → ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
47		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
48	47	fmpt 7134	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵)
49	46, 48	sylib 221	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵)
50		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵
51	50	elrabsf 3787	. . . . . . . . . 10 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵 ∧ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
52	51	simprbi 503	. . . . . . . . 9 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
53	52	ralimi 3103	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
54	53	ad2antll 742	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})) → ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
55	49, 54	jca 521	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})) → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
56		feq1 6709	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑓:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵))
57		nfmpt1 5207	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
58	57	nfeq2 2943	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
59		fvex 6921	. . . . . . . . . 10 ⊢ (𝑓‘𝑥) ∈ V
60		ac6num.1	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))
61	59, 60	sbcie 3783	. . . . . . . . 9 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ 𝜓)
62		fveq1 6907	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑓‘𝑥) = ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥))
63		fvex 6921	. . . . . . . . . . . 12 ⊢ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ V
64	47	fvmpt2 7028	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ V) → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
65	63, 64	mpan2 704	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
66	62, 65	sylan9eq 2818	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
67	66	sbceq1d 3747	. . . . . . . . 9 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
68	61, 67	bitr3id 288	. . . . . . . 8 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
69	58, 68	ralbida 3277	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
70	56, 69	anbi12d 644	. . . . . 6 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → ((𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓) ↔ ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)))
71	43, 55, 70	spcedv 3556	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑})) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
72	71	ex 418	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ((𝑔 Fn ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
73	41, 72	syld 48	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ((𝑔:ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})⟶∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
74	73	exlimdv 1968	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → (∃𝑔(𝑔:ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})⟶∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})(𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
75	31, 74	mpd 16	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454  [wsbc 3742   ⊆ wss 3900  ∅c0 4281  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5189  dom cdm 5655  ran crn 5656   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  cardccrd 10043

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-en 9002  df-card 10047

This theorem is used by:  ac6  10585  ptcmplem3  24403  poimirlem32  38762