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Theorem ac6num 9554
Description: A version of ac6 9555 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 4706 . . . . . . . . 9 𝑥 𝑥𝐴 {𝑦𝐵𝜑}
21nfel1 2922 . . . . . . . 8 𝑥 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card
3 ssiun2 4719 . . . . . . . . 9 (𝑥𝐴 → {𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑})
4 ssexg 4965 . . . . . . . . . 10 (({𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑} ∧ 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card) → {𝑦𝐵𝜑} ∈ V)
54expcom 402 . . . . . . . . 9 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ({𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑} → {𝑦𝐵𝜑} ∈ V))
63, 5syl5 34 . . . . . . . 8 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → (𝑥𝐴 → {𝑦𝐵𝜑} ∈ V))
72, 6ralrimi 3104 . . . . . . 7 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
8 dfiun2g 4708 . . . . . . 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 2765 . . . . . . . 8 (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = (𝑥𝐴 ↦ {𝑦𝐵𝜑})
1110rnmpt 5540 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
1211unieqi 4603 . . . . . 6 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
139, 12syl6eqr 2817 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} = ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
14 id 22 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
1513, 14eqeltrrd 2845 . . . 4 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
16153ad2ant2 1164 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
17 simp3 1168 . . . . 5 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴𝑦𝐵 𝜑)
18 necom 2990 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦𝐵𝜑})
19 rabn0 4122 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∃𝑦𝐵 𝜑)
20 df-ne 2938 . . . . . . . 8 (∅ ≠ {𝑦𝐵𝜑} ↔ ¬ ∅ = {𝑦𝐵𝜑})
2118, 19, 203bitr3i 292 . . . . . . 7 (∃𝑦𝐵 𝜑 ↔ ¬ ∅ = {𝑦𝐵𝜑})
2221ralbii 3127 . . . . . 6 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑})
23 ralnex 3139 . . . . . 6 (∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑} ↔ ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2422, 23bitri 266 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2517, 24sylib 209 . . . 4 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
26 0ex 4950 . . . . 5 ∅ ∈ V
2710elrnmpt 5541 . . . . 5 (∅ ∈ V → (∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ↔ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑}))
2826, 27ax-mp 5 . . . 4 (∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ↔ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2925, 28sylnibr 320 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ¬ ∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
30 ac5num 9110 . . 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 579 . 2 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑔(𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧))
32 ffn 6223 . . . . . 6 (𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) → 𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
3332anim1i 608 . . . . 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 1164 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
35 fveq2 6375 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → (𝑔𝑧) = (𝑔‘{𝑦𝐵𝜑}))
36 id 22 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → 𝑧 = {𝑦𝐵𝜑})
3735, 36eleq12d 2838 . . . . . . . 8 (𝑧 = {𝑦𝐵𝜑} → ((𝑔𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}))
3810, 37ralrnmpt 6558 . . . . . . 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 622 . . . . 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 235 . . . 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 3760 . . . . . . . . . . 11 (𝑥𝐴 → ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}))
4342ralimia 3097 . . . . . . . . . 10 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
4443ad2antll 720 . . . . . . . . 9 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
45 nfv 2009 . . . . . . . . . 10 𝑧(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}
46 nfcsb1v 3707 . . . . . . . . . . 11 𝑥𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑})
4746, 1nfel 2920 . . . . . . . . . 10 𝑥𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}
48 csbeq1a 3700 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑔‘{𝑦𝐵𝜑}) = 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}))
4948eleq1d 2829 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑} ↔ 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑}))
5045, 47, 49cbvral 3315 . . . . . . . . 9 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑} ↔ ∀𝑧𝐴 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑})
5144, 50sylib 209 . . . . . . . 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 2907 . . . . . . . . . 10 𝑧(𝑔‘{𝑦𝐵𝜑})
5352, 46, 48cbvmpt 4908 . . . . . . . . 9 (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) = (𝑧𝐴𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}))
5453fmpt 6570 . . . . . . . 8 (∀𝑧𝐴 𝑧 / 𝑥(𝑔‘{𝑦𝐵𝜑}) ∈ 𝑥𝐴 {𝑦𝐵𝜑} ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴 𝑥𝐴 {𝑦𝐵𝜑})
5551, 54sylib 209 . . . . . . 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 1242 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → 𝐴𝑉)
57 simpl2 1244 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
58 fex2 7319 . . . . . . 7 (((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴 𝑥𝐴 {𝑦𝐵𝜑} ∧ 𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V)
5955, 56, 57, 58syl3anc 1490 . . . . . 6 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V)
60 ssrab2 3847 . . . . . . . . . . 11 {𝑦𝐵𝜑} ⊆ 𝐵
6160sseli 3757 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
6261ralimi 3099 . . . . . . . . 9 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
6362ad2antll 720 . . . . . . . 8 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
64 eqid 2765 . . . . . . . . 9 (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
6564fmpt 6570 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵)
6663, 65sylib 209 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵)
67 nfcv 2907 . . . . . . . . . . 11 𝑦𝐵
6867elrabsf 3635 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} ↔ ((𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
6968simprbi 490 . . . . . . . . 9 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7069ralimi 3099 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7170ad2antll 720 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
7266, 71jca 507 . . . . . 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 6204 . . . . . . . 8 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵))
74 nfmpt1 4906 . . . . . . . . . 10 𝑥(𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
7574nfeq2 2923 . . . . . . . . 9 𝑥 𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
76 fvex 6388 . . . . . . . . . . 11 (𝑓𝑥) ∈ V
77 ac6num.1 . . . . . . . . . . 11 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
7876, 77sbcie 3631 . . . . . . . . . 10 ([(𝑓𝑥) / 𝑦]𝜑𝜓)
79 fveq1 6374 . . . . . . . . . . . 12 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓𝑥) = ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥))
80 fvex 6388 . . . . . . . . . . . . 13 (𝑔‘{𝑦𝐵𝜑}) ∈ V
8164fvmpt2 6480 . . . . . . . . . . . . 13 ((𝑥𝐴 ∧ (𝑔‘{𝑦𝐵𝜑}) ∈ V) → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8280, 81mpan2 682 . . . . . . . . . . . 12 (𝑥𝐴 → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8379, 82sylan9eq 2819 . . . . . . . . . . 11 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝑓𝑥) = (𝑔‘{𝑦𝐵𝜑}))
8483sbceq1d 3601 . . . . . . . . . 10 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → ([(𝑓𝑥) / 𝑦]𝜑[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
8578, 84syl5bbr 276 . . . . . . . . 9 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝜓[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
8675, 85ralbida 3129 . . . . . . . 8 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
8773, 86anbi12d 624 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓) ↔ ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)))
8887spcegv 3446 . . . . . 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 401 . . . 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 2028 . 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  [wsbc 3596  csb 3691  wss 3732  c0 4079   cuni 4594   ciun 4676  cmpt 4888  dom cdm 5277  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-en 8161  df-card 9016
This theorem is referenced by:  ac6  9555  ptcmplem3  22137  poimirlem32  33797
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