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Theorem ac6num 9493
Description: A version of ac6 9494 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 4702 . . . . . . . . 9 𝑥 𝑥𝐴 {𝑦𝐵𝜑}
21nfel1 2917 . . . . . . . 8 𝑥 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card
3 ssiun2 4715 . . . . . . . . 9 (𝑥𝐴 → {𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑})
4 ssexg 4956 . . . . . . . . . 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 3095 . . . . . . 7 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
8 dfiun2g 4704 . . . . . . 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 2760 . . . . . . . 8 (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = (𝑥𝐴 ↦ {𝑦𝐵𝜑})
1110rnmpt 5526 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
1211unieqi 4597 . . . . . 6 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
139, 12syl6eqr 2812 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} = ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
14 id 22 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
1513, 14eqeltrrd 2840 . . . 4 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
16153ad2ant2 1129 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
17 simp3 1133 . . . . 5 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴𝑦𝐵 𝜑)
18 necom 2985 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦𝐵𝜑})
19 rabn0 4101 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∃𝑦𝐵 𝜑)
20 df-ne 2933 . . . . . . . 8 (∅ ≠ {𝑦𝐵𝜑} ↔ ¬ ∅ = {𝑦𝐵𝜑})
2118, 19, 203bitr3i 290 . . . . . . 7 (∃𝑦𝐵 𝜑 ↔ ¬ ∅ = {𝑦𝐵𝜑})
2221ralbii 3118 . . . . . 6 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑})
23 ralnex 3130 . . . . . 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 4942 . . . . 5 ∅ ∈ V
2710elrnmpt 5527 . . . . 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 9049 . . 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 696 . 2 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑔(𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧))
32 ffn 6206 . . . . . 6 (𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) → 𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
3332anim1i 593 . . . . 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 1129 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
35 fveq2 6352 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → (𝑔𝑧) = (𝑔‘{𝑦𝐵𝜑}))
36 id 22 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → 𝑧 = {𝑦𝐵𝜑})
3735, 36eleq12d 2833 . . . . . . . 8 (𝑧 = {𝑦𝐵𝜑} → ((𝑔𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}))
3810, 37ralrnmpt 6531 . . . . . . 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 742 . . . . 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 3743 . . . . . . . . . . 11 (𝑥𝐴 → ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}))
4342ralimia 3088 . . . . . . . . . 10 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
4443ad2antll 767 . . . . . . . . 9 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
45 nfv 1992 . . . . . . . . . 10 𝑧(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}
46 nfcsb1v 3690 . . . . . . . . . . 11 𝑥𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑})
4746, 1nfel 2915 . . . . . . . . . 10 𝑥𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}
48 csbeq1a 3683 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑔‘{𝑦𝐵𝜑}) = 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}))
4948eleq1d 2824 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑} ↔ 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}))
5045, 47, 49cbvral 3306 . . . . . . . . 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 2902 . . . . . . . . . 10 𝑧(𝑔‘{𝑦𝐵𝜑})
5352, 46, 48cbvmpt 4901 . . . . . . . . 9 (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) = (𝑧𝐴𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}))
5453fmpt 6544 . . . . . . . 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 1228 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → 𝐴𝑉)
57 simpl2 1230 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
58 fex2 7286 . . . . . . 7 (((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴 𝑥𝐴 {𝑦𝐵𝜑} ∧ 𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V)
5955, 56, 57, 58syl3anc 1477 . . . . . 6 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V)
60 ssrab2 3828 . . . . . . . . . . 11 {𝑦𝐵𝜑} ⊆ 𝐵
6160sseli 3740 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
6261ralimi 3090 . . . . . . . . 9 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
6362ad2antll 767 . . . . . . . 8 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
64 eqid 2760 . . . . . . . . 9 (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
6564fmpt 6544 . . . . . . . 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 2902 . . . . . . . . . . 11 𝑦𝐵
6867elrabsf 3615 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} ↔ ((𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
6968simprbi 483 . . . . . . . . 9 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7069ralimi 3090 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7170ad2antll 767 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7266, 71jca 555 . . . . . 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 6187 . . . . . . . 8 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵))
74 nfmpt1 4899 . . . . . . . . . 10 𝑥(𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
7574nfeq2 2918 . . . . . . . . 9 𝑥 𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
76 fvex 6362 . . . . . . . . . . 11 (𝑓𝑥) ∈ V
77 ac6num.1 . . . . . . . . . . 11 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
7876, 77sbcie 3611 . . . . . . . . . 10 ([(𝑓𝑥) / 𝑦]𝜑𝜓)
79 fveq1 6351 . . . . . . . . . . . 12 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓𝑥) = ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥))
80 fvex 6362 . . . . . . . . . . . . 13 (𝑔‘{𝑦𝐵𝜑}) ∈ V
8164fvmpt2 6453 . . . . . . . . . . . . 13 ((𝑥𝐴 ∧ (𝑔‘{𝑦𝐵𝜑}) ∈ V) → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8280, 81mpan2 709 . . . . . . . . . . . 12 (𝑥𝐴 → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8379, 82sylan9eq 2814 . . . . . . . . . . 11 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝑓𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8483sbceq1d 3581 . . . . . . . . . 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 3120 . . . . . . . 8 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
8773, 86anbi12d 749 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓) ↔ ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)))
8887spcegv 3434 . . . . . 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 2010 . 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 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  [wsbc 3576  csb 3674  wss 3715  c0 4058   cuni 4588   ciun 4672  cmpt 4881  dom cdm 5266  ran crn 5267   Fn wfn 6044  wf 6045  cfv 6049  cardccrd 8951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-en 8122  df-card 8955
This theorem is referenced by:  ac6  9494  ptcmplem3  22059  poimirlem32  33754
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