Theorem ac6num 10401

Description: A version of ac6 10402 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 4969	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑}
2	1	nfel1 2915	. . . . . . . 8 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card
3		ssiun2 4990	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
4		ssexg 5264	. . . . . . . . . 10 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card) → {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
5	4	expcom 413	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ({𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} → {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V))
6	3, 5	syl5 34	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V))
7	2, 6	ralrimi 3235	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
8		dfiun2g 4972	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}})
9	7, 8	syl 17	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}})
10		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})
11	10	rnmpt 5912	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}}
12	11	unieqi 4862	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}}
13	9, 12	eqtr4di 2789	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}))
14		id 22	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card)
15	13, 14	eqeltrrd 2837	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ dom card)
16	15	3ad2ant2 1135	. . 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 1139	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
18		necom 2985	. . . . . . . 8 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦 ∈ 𝐵 ∣ 𝜑})
19		rabn0 4329	. . . . . . . 8 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
20		df-ne 2933	. . . . . . . 8 ⊢ (∅ ≠ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
21	18, 19, 20	3bitr3i 301	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
22	21	ralbii 3083	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
23		ralnex 3063	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
24	22, 23	bitri 275	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
25	17, 24	sylib 218	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
26		0ex 5242	. . . . 5 ⊢ ∅ ∈ V
27	10	elrnmpt 5913	. . . . 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 329	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ¬ ∅ ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}))
30		ac5num 9958	. . 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 585	. 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 6668	. . . . . 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 616	. . . . 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 1135	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
35		fveq2 6840	. . . . . . . . 9 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑔‘𝑧) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
36		id 22	. . . . . . . . 9 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑})
37	35, 36	eleq12d 2830	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ((𝑔‘𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}))
38	10, 37	ralrnmptw 7046	. . . . . . 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 17	. . . . . 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 631	. . . . 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 244	. . . 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 1193	. . . . . . 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 7179	. . . . . 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 3630	. . . . . . . . . 10 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
45	44	ralimi 3074	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
46	45	ad2antll 730	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
48	47	fmpt 7062	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵)
49	46, 48	sylib 218	. . . . . . 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 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵
51	50	elrabsf 3774	. . . . . . . . . 10 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵 ∧ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
52	51	simprbi 497	. . . . . . . . 9 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
53	52	ralimi 3074	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
54	53	ad2antll 730	. . . . . . 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 511	. . . . . 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 6646	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑓:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵))
57		nfmpt1 5184	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
58	57	nfeq2 2916	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
59		fvex 6853	. . . . . . . . . 10 ⊢ (𝑓‘𝑥) ∈ V
60		ac6num.1	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))
61	59, 60	sbcie 3770	. . . . . . . . 9 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ 𝜓)
62		fveq1 6839	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑓‘𝑥) = ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥))
63		fvex 6853	. . . . . . . . . . . 12 ⊢ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ V
64	47	fvmpt2 6959	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ V) → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
65	63, 64	mpan2 692	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
66	62, 65	sylan9eq 2791	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
67	66	sbceq1d 3733	. . . . . . . . 9 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
68	61, 67	bitr3id 285	. . . . . . . 8 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
69	58, 68	ralbida 3248	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
70	56, 69	anbi12d 633	. . . . . 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 3540	. . . . 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 412	. . . 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 47	. . 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 1935	. 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 15	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  {crab 3389  Vcvv 3429  [wsbc 3728   ⊆ wss 3889  ∅c0 4273  ∪ cuni 4850  ∪ ciun 4933   ↦ cmpt 5166  dom cdm 5631  ran crn 5632   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-en 8894  df-card 9863

This theorem is referenced by:  ac6  10402  ptcmplem3  24019  poimirlem32  37973