Theorem ac6num 10362

Description: A version of ac6 10363 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 4975	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑}
2	1	nfel1 2909	. . . . . . . 8 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card
3		ssiun2 4994	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑})
4		ssexg 5259	. . . . . . . . . 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 3228	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
8		dfiun2g 4978	. . . . . . 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 2730	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})
11	10	rnmpt 5894	. . . . . . 7 ⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}}
12	11	unieqi 4869	. . . . . 6 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑}}
13	9, 12	eqtr4di 2783	. . . . 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 2830	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card → ∪ ran (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ dom card)
16	15	3ad2ant2 1134	. . 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 1138	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
18		necom 2979	. . . . . . . 8 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦 ∈ 𝐵 ∣ 𝜑})
19		rabn0 4337	. . . . . . . 8 ⊢ ({𝑦 ∈ 𝐵 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝐵 𝜑)
20		df-ne 2927	. . . . . . . 8 ⊢ (∅ ≠ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
21	18, 19, 20	3bitr3i 301	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
22	21	ralbii 3076	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ ∅ = {𝑦 ∈ 𝐵 ∣ 𝜑})
23		ralnex 3056	. . . . . 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 5243	. . . . 5 ⊢ ∅ ∈ V
27	10	elrnmpt 5895	. . . . 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 9919	. . 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 584	. 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 6647	. . . . . 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 615	. . . . 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 1134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
35		fveq2 6817	. . . . . . . . 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 2823	. . . . . . . 8 ⊢ (𝑧 = {𝑦 ∈ 𝐵 ∣ 𝜑} → ((𝑔‘𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑}))
38	10, 37	ralrnmptw 7022	. . . . . . 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 630	. . . . 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 1192	. . . . . . 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 7153	. . . . . 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 3641	. . . . . . . . . 10 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
45	44	ralimi 3067	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵)
46	45	ad2antll 729	. . . . . . . 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 2730	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
48	47	fmpt 7038	. . . . . . . 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 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵
51	50	elrabsf 3785	. . . . . . . . . 10 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} ↔ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ 𝐵 ∧ [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
52	51	simprbi 496	. . . . . . . . 9 ⊢ ((𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
53	52	ralimi 3067	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ {𝑦 ∈ 𝐵 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑)
54	53	ad2antll 729	. . . . . . 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 6625	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑓:𝐴⟶𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})):𝐴⟶𝐵))
57		nfmpt1 5188	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
58	57	nfeq2 2910	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
59		fvex 6830	. . . . . . . . . 10 ⊢ (𝑓‘𝑥) ∈ V
60		ac6num.1	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))
61	59, 60	sbcie 3781	. . . . . . . . 9 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ 𝜓)
62		fveq1 6816	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (𝑓‘𝑥) = ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥))
63		fvex 6830	. . . . . . . . . . . 12 ⊢ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ V
64	47	fvmpt2 6935	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) ∈ V) → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
65	63, 64	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
66	62, 65	sylan9eq 2785	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}))
67	66	sbceq1d 3744	. . . . . . . . 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 3241	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑})) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 [(𝑔‘{𝑦 ∈ 𝐵 ∣ 𝜑}) / 𝑦]𝜑))
70	56, 69	anbi12d 632	. . . . . 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 3551	. . . . 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 1934	. 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 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2110  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  {crab 3393  Vcvv 3434  [wsbc 3739   ⊆ wss 3900  ∅c0 4281  ∪ cuni 4857  ∪ ciun 4939   ↦ cmpt 5170  dom cdm 5614  ran crn 5615   Fn wfn 6472  ⟶wf 6473  ‘cfv 6477  cardccrd 9820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6305  df-on 6306  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-en 8865  df-card 9824

This theorem is referenced by:  ac6  10363  ptcmplem3  23962  poimirlem32  37671