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Theorem ac6num 9893
Description: A version of ac6 9894 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 4944 . . . . . . . . 9 𝑥 𝑥𝐴 {𝑦𝐵𝜑}
21nfel1 2992 . . . . . . . 8 𝑥 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card
3 ssiun2 4962 . . . . . . . . 9 (𝑥𝐴 → {𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑})
4 ssexg 5218 . . . . . . . . . 10 (({𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑} ∧ 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card) → {𝑦𝐵𝜑} ∈ V)
54expcom 416 . . . . . . . . 9 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ({𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑} → {𝑦𝐵𝜑} ∈ V))
63, 5syl5 34 . . . . . . . 8 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → (𝑥𝐴 → {𝑦𝐵𝜑} ∈ V))
72, 6ralrimi 3214 . . . . . . 7 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
8 dfiun2g 4946 . . . . . . 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 2819 . . . . . . . 8 (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = (𝑥𝐴 ↦ {𝑦𝐵𝜑})
1110rnmpt 5820 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
1211unieqi 4839 . . . . . 6 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
139, 12syl6eqr 2872 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} = ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
14 id 22 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
1513, 14eqeltrrd 2912 . . . 4 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
16153ad2ant2 1128 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
17 simp3 1132 . . . . 5 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴𝑦𝐵 𝜑)
18 necom 3067 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦𝐵𝜑})
19 rabn0 4337 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∃𝑦𝐵 𝜑)
20 df-ne 3015 . . . . . . . 8 (∅ ≠ {𝑦𝐵𝜑} ↔ ¬ ∅ = {𝑦𝐵𝜑})
2118, 19, 203bitr3i 303 . . . . . . 7 (∃𝑦𝐵 𝜑 ↔ ¬ ∅ = {𝑦𝐵𝜑})
2221ralbii 3163 . . . . . 6 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑})
23 ralnex 3234 . . . . . 6 (∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑} ↔ ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2422, 23bitri 277 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2517, 24sylib 220 . . . 4 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
26 0ex 5202 . . . . 5 ∅ ∈ V
2710elrnmpt 5821 . . . . 5 (∅ ∈ V → (∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ↔ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑}))
2826, 27ax-mp 5 . . . 4 (∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ↔ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2925, 28sylnibr 331 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ¬ ∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
30 ac5num 9454 . . 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 586 . 2 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑔(𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧))
32 ffn 6507 . . . . . 6 (𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) → 𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
3332anim1i 616 . . . . 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 1128 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
35 fveq2 6663 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → (𝑔𝑧) = (𝑔‘{𝑦𝐵𝜑}))
36 id 22 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → 𝑧 = {𝑦𝐵𝜑})
3735, 36eleq12d 2905 . . . . . . . 8 (𝑧 = {𝑦𝐵𝜑} → ((𝑔𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}))
3810, 37ralrnmptw 6853 . . . . . . 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 630 . . . . 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 246 . . . 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})))
42 simpl1 1185 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → 𝐴𝑉)
4342mptexd 6979 . . . . . 6 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V)
44 ssrab2 4054 . . . . . . . . . . 11 {𝑦𝐵𝜑} ⊆ 𝐵
4544sseli 3961 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
4645ralimi 3158 . . . . . . . . 9 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
4746ad2antll 727 . . . . . . . 8 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
48 eqid 2819 . . . . . . . . 9 (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
4948fmpt 6867 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵)
5047, 49sylib 220 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵)
51 nfcv 2975 . . . . . . . . . . 11 𝑦𝐵
5251elrabsf 3814 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} ↔ ((𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
5352simprbi 499 . . . . . . . . 9 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
5453ralimi 3158 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
5554ad2antll 727 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
5650, 55jca 514 . . . . . 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))
57 feq1 6488 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵))
58 nfmpt1 5155 . . . . . . . . 9 𝑥(𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
5958nfeq2 2993 . . . . . . . 8 𝑥 𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
60 fvex 6676 . . . . . . . . . 10 (𝑓𝑥) ∈ V
61 ac6num.1 . . . . . . . . . 10 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
6260, 61sbcie 3810 . . . . . . . . 9 ([(𝑓𝑥) / 𝑦]𝜑𝜓)
63 fveq1 6662 . . . . . . . . . . 11 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓𝑥) = ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥))
64 fvex 6676 . . . . . . . . . . . 12 (𝑔‘{𝑦𝐵𝜑}) ∈ V
6548fvmpt2 6772 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ (𝑔‘{𝑦𝐵𝜑}) ∈ V) → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
6664, 65mpan2 689 . . . . . . . . . . 11 (𝑥𝐴 → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
6763, 66sylan9eq 2874 . . . . . . . . . 10 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝑓𝑥) = (𝑔‘{𝑦𝐵𝜑}))
6867sbceq1d 3775 . . . . . . . . 9 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → ([(𝑓𝑥) / 𝑦]𝜑[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
6962, 68syl5bbr 287 . . . . . . . 8 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝜓[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
7059, 69ralbida 3228 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
7157, 70anbi12d 632 . . . . . 6 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓) ↔ ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)))
7243, 56, 71spcedv 3597 . . . . 5 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
7372ex 415 . . . 4 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ((𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
7441, 73syld 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)))
7574exlimdv 1927 . 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)))
7631, 75mpd 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 208  wa 398  w3a 1081   = wceq 1530  wex 1773  wcel 2107  {cab 2797  wne 3014  wral 3136  wrex 3137  {crab 3140  Vcvv 3493  [wsbc 3770  wss 3934  c0 4289   cuni 4830   ciun 4910  cmpt 5137  dom cdm 5548  ran crn 5549   Fn wfn 6343  wf 6344  cfv 6348  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-en 8502  df-card 9360
This theorem is referenced by:  ac6  9894  ptcmplem3  22654  poimirlem32  34911
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