Theorem ac6num 9301

Description: A version of ac6 9302 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 4550	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑}
2	1	nfel1 2779	. . . . . . . 8 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card
3		ssiun2 4563	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
4		ssexg 4804	. . . . . . . . . 10 ⊢ (({𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card) → {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
5	4	expcom 451	. . . . . . . . 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 2957	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
8		dfiun2g 4552	. . . . . . 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 2622	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})
11	10	rnmpt 5371	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}}
12	11	unieqi 4445	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}}
13	9, 12	syl6eqr 2674	. . . . 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 2702	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ dom card)
16	15	3ad2ant2 1083	. . 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 1063	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
18		necom 2847	. . . . . . . 8 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦 ∈ 𝐵 ∣ 𝜑})
19		rabn0 3958	. . . . . . . 8 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
20		df-ne 2795	. . . . . . . 8 ⊢ (∅ ≠ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
21	18, 19, 20	3bitr3i 290	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
22	21	ralbii 2980	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
23		ralnex 2992	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
24	22, 23	bitri 264	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
25	17, 24	sylib 208	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ¬ ∃𝑥 ∈ 𝐴 ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
26		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
27	10	elrnmpt 5372	. . . . 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 319	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ¬ ∅ ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}))
30		ac5num 8859	. . 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 693	. 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 6045	. . . . . 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 592	. . . . 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 1083	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
35		fveq2 6191	. . . . . . . . 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 2695	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ((𝑔‘𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}))
38	10, 37	ralrnmpt 6368	. . . . . . 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 740	. . . . 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	syl5ib 234	. . . 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	3	sseld 3602	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑}))
43	42	ralimia 2950	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
44	43	ad2antll 765	. . . . . . . . 9 ⊢ (((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}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
45		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑}
46		nfcsb1v 3549	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})
47	46, 1	nfel 2777	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑}
48		csbeq1a 3542	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) = ⦋𝑧 / 𝑥⦌(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
49	48	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ⦋𝑧 / 𝑥⦌(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑}))
50	45, 47, 49	cbvral 3167	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑥⦌(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
51	44, 50	sylib 208	. . . . . . . 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})) → ∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑥⦌(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
52		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})
53	52, 46, 48	cbvmpt 4749	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
54	53	fmpt 6381	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ⦋𝑧 / 𝑥⦌(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
55	51, 54	sylib 208	. . . . . . 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⟶∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
56		simpl1 1064	. . . . . . 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)
57		simpl2 1065	. . . . . . 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 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card)
58		fex2 7121	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∧ 𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card) → (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∈ V)
59	55, 56, 57, 58	syl3anc 1326	. . . . . 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)
60		ssrab2 3687	. . . . . . . . . . 11 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵
61	60	sseli 3599	. . . . . . . . . 10 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
62	61	ralimi 2952	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
63	62	ad2antll 765	. . . . . . . 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)
64		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
65	64	fmpt 6381	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵)
66	63, 65	sylib 208	. . . . . . 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)
67		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵
68	67	elrabsf 3474	. . . . . . . . . 10 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵 ∧ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
69	68	simprbi 480	. . . . . . . . 9 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
70	69	ralimi 2952	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
71	70	ad2antll 765	. . . . . . 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)
72	66, 71	jca 554	. . . . . 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))
73		feq1 6026	. . . . . . . 8 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑓:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵))
74		nfmpt1 4747	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
75	74	nfeq2 2780	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
76		fvex 6201	. . . . . . . . . . 11 ⊢ (𝑓‘𝑥) ∈ V
77		ac6num.1	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))
78	76, 77	sbcie 3470	. . . . . . . . . 10 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ 𝜓)
79		fveq1 6190	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑓‘𝑥) = ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥))
80		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ V
81	64	fvmpt2 6291	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ V) → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
82	80, 81	mpan2 707	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
83	79, 82	sylan9eq 2676	. . . . . . . . . . 11 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
84	83	sbceq1d 3440	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
85	78, 84	syl5bbr 274	. . . . . . . . 9 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
86	75, 85	ralbida 2982	. . . . . . . 8 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
87	73, 86	anbi12d 747	. . . . . . 7 ⊢ (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)))
88	87	spcegv 3294	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∈ V → (((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
89	59, 72, 88	sylc 65	. . . . 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))
90	89	ex 450	. . . 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)))
91	41, 90	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)))
92	91	exlimdv 1861	. 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)))
93	31, 92	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200  [wsbc 3435  ⦋csb 3533   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436  ∪ ciun 4520   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-en 7956  df-card 8765

This theorem is referenced by:  ac6  9302  ptcmplem3  21858  poimirlem32  33441