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Theorem ac6num 9339
 Description: A version of ac6 9340 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
Hypothesis
Ref Expression
ac6num.1 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
Assertion
Ref Expression
ac6num ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 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.
StepHypRef Expression
1 nfiu1 4582 . . . . . . . . 9 𝑥 𝑥𝐴 {𝑦𝐵𝜑}
21nfel1 2808 . . . . . . . 8 𝑥 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card
3 ssiun2 4595 . . . . . . . . 9 (𝑥𝐴 → {𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑})
4 ssexg 4837 . . . . . . . . . 10 (({𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑} ∧ 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card) → {𝑦𝐵𝜑} ∈ V)
54expcom 450 . . . . . . . . 9 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ({𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑} → {𝑦𝐵𝜑} ∈ V))
63, 5syl5 34 . . . . . . . 8 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → (𝑥𝐴 → {𝑦𝐵𝜑} ∈ V))
72, 6ralrimi 2986 . . . . . . 7 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
8 dfiun2g 4584 . . . . . . 7 (∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}})
97, 8syl 17 . . . . . 6 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}})
10 eqid 2651 . . . . . . . 8 (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = (𝑥𝐴 ↦ {𝑦𝐵𝜑})
1110rnmpt 5403 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
1211unieqi 4477 . . . . . 6 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
139, 12syl6eqr 2703 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} = ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
14 id 22 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
1513, 14eqeltrrd 2731 . . . 4 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
16153ad2ant2 1103 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
17 simp3 1083 . . . . 5 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴𝑦𝐵 𝜑)
18 necom 2876 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦𝐵𝜑})
19 rabn0 3991 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∃𝑦𝐵 𝜑)
20 df-ne 2824 . . . . . . . 8 (∅ ≠ {𝑦𝐵𝜑} ↔ ¬ ∅ = {𝑦𝐵𝜑})
2118, 19, 203bitr3i 290 . . . . . . 7 (∃𝑦𝐵 𝜑 ↔ ¬ ∅ = {𝑦𝐵𝜑})
2221ralbii 3009 . . . . . 6 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑})
23 ralnex 3021 . . . . . 6 (∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑} ↔ ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2422, 23bitri 264 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2517, 24sylib 208 . . . 4 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
26 0ex 4823 . . . . 5 ∅ ∈ V
2710elrnmpt 5404 . . . . 5 (∅ ∈ V → (∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ↔ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑}))
2826, 27ax-mp 5 . . . 4 (∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ↔ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2925, 28sylnibr 318 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ¬ ∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
30 ac5num 8897 . . 3 (( ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card ∧ ¬ ∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})) → ∃𝑔(𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧))
3116, 29, 30syl2anc 694 . 2 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑔(𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧))
32 ffn 6083 . . . . . 6 (𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) → 𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
3332anim1i 591 . . . . 5 ((𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧) → (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧))
3473ad2ant2 1103 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
35 fveq2 6229 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → (𝑔𝑧) = (𝑔‘{𝑦𝐵𝜑}))
36 id 22 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → 𝑧 = {𝑦𝐵𝜑})
3735, 36eleq12d 2724 . . . . . . . 8 (𝑧 = {𝑦𝐵𝜑} → ((𝑔𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}))
3810, 37ralrnmpt 6408 . . . . . . 7 (∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V → (∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧 ↔ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}))
3934, 38syl 17 . . . . . 6 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → (∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧 ↔ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}))
4039anbi2d 740 . . . . 5 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ((𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧) ↔ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})))
4133, 40syl5ib 234 . . . 4 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ((𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧) → (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})))
423sseld 3635 . . . . . . . . . . 11 (𝑥𝐴 → ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}))
4342ralimia 2979 . . . . . . . . . 10 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
4443ad2antll 765 . . . . . . . . 9 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
45 nfv 1883 . . . . . . . . . 10 𝑧(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}
46 nfcsb1v 3582 . . . . . . . . . . 11 𝑥𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑})
4746, 1nfel 2806 . . . . . . . . . 10 𝑥𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}
48 csbeq1a 3575 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑔‘{𝑦𝐵𝜑}) = 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}))
4948eleq1d 2715 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑} ↔ 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}))
5045, 47, 49cbvral 3197 . . . . . . . . 9 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑} ↔ ∀𝑧𝐴 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
5144, 50sylib 208 . . . . . . . 8 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑧𝐴 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
52 nfcv 2793 . . . . . . . . . 10 𝑧(𝑔‘{𝑦𝐵𝜑})
5352, 46, 48cbvmpt 4782 . . . . . . . . 9 (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) = (𝑧𝐴𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}))
5453fmpt 6421 . . . . . . . 8 (∀𝑧𝐴 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑} ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴 𝑥𝐴 {𝑦𝐵𝜑})
5551, 54sylib 208 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴 𝑥𝐴 {𝑦𝐵𝜑})
56 simpl1 1084 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → 𝐴𝑉)
57 simpl2 1085 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
58 fex2 7163 . . . . . . 7 (((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴 𝑥𝐴 {𝑦𝐵𝜑} ∧ 𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V)
5955, 56, 57, 58syl3anc 1366 . . . . . 6 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V)
60 ssrab2 3720 . . . . . . . . . . 11 {𝑦𝐵𝜑} ⊆ 𝐵
6160sseli 3632 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
6261ralimi 2981 . . . . . . . . 9 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
6362ad2antll 765 . . . . . . . 8 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
64 eqid 2651 . . . . . . . . 9 (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
6564fmpt 6421 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵)
6663, 65sylib 208 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵)
67 nfcv 2793 . . . . . . . . . . 11 𝑦𝐵
6867elrabsf 3507 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} ↔ ((𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
6968simprbi 479 . . . . . . . . 9 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7069ralimi 2981 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7170ad2antll 765 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7266, 71jca 553 . . . . . 6 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
73 feq1 6064 . . . . . . . 8 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵))
74 nfmpt1 4780 . . . . . . . . . 10 𝑥(𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
7574nfeq2 2809 . . . . . . . . 9 𝑥 𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
76 fvex 6239 . . . . . . . . . . 11 (𝑓𝑥) ∈ V
77 ac6num.1 . . . . . . . . . . 11 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
7876, 77sbcie 3503 . . . . . . . . . 10 ([(𝑓𝑥) / 𝑦]𝜑𝜓)
79 fveq1 6228 . . . . . . . . . . . 12 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓𝑥) = ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥))
80 fvex 6239 . . . . . . . . . . . . 13 (𝑔‘{𝑦𝐵𝜑}) ∈ V
8164fvmpt2 6330 . . . . . . . . . . . . 13 ((𝑥𝐴 ∧ (𝑔‘{𝑦𝐵𝜑}) ∈ V) → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8280, 81mpan2 707 . . . . . . . . . . . 12 (𝑥𝐴 → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8379, 82sylan9eq 2705 . . . . . . . . . . 11 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝑓𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8483sbceq1d 3473 . . . . . . . . . 10 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → ([(𝑓𝑥) / 𝑦]𝜑[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
8578, 84syl5bbr 274 . . . . . . . . 9 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝜓[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
8675, 85ralbida 3011 . . . . . . . 8 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
8773, 86anbi12d 747 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓) ↔ ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)))
8887spcegv 3325 . . . . . 6 ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V → (((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
8959, 72, 88sylc 65 . . . . 5 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
9089ex 449 . . . 4 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ((𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
9141, 90syld 47 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ((𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
9291exlimdv 1901 . 2 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → (∃𝑔(𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
9331, 92mpd 15 1 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231  [wsbc 3468  ⦋csb 3566   ⊆ wss 3607  ∅c0 3948  ∪ cuni 4468  ∪ ciun 4552   ↦ cmpt 4762  dom cdm 5143  ran crn 5144   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  cardccrd 8799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-en 7998  df-card 8803 This theorem is referenced by:  ac6  9340  ptcmplem3  21905  poimirlem32  33571
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