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Theorem ac6num 10470
Description: A version of ac6 10471 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 5030 . . . . . . . . 9 𝑥 𝑥𝐴 {𝑦𝐵𝜑}
21nfel1 2920 . . . . . . . 8 𝑥 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card
3 ssiun2 5049 . . . . . . . . 9 (𝑥𝐴 → {𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑})
4 ssexg 5322 . . . . . . . . . 10 (({𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑} ∧ 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card) → {𝑦𝐵𝜑} ∈ V)
54expcom 415 . . . . . . . . 9 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ({𝑦𝐵𝜑} ⊆ 𝑥𝐴 {𝑦𝐵𝜑} → {𝑦𝐵𝜑} ∈ V))
63, 5syl5 34 . . . . . . . 8 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → (𝑥𝐴 → {𝑦𝐵𝜑} ∈ V))
72, 6ralrimi 3255 . . . . . . 7 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
8 dfiun2g 5032 . . . . . . 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 2733 . . . . . . . 8 (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = (𝑥𝐴 ↦ {𝑦𝐵𝜑})
1110rnmpt 5952 . . . . . . 7 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
1211unieqi 4920 . . . . . 6 ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = {𝑦𝐵𝜑}}
139, 12eqtr4di 2791 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} = ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
14 id 22 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
1513, 14eqeltrrd 2835 . . . 4 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
16153ad2ant2 1135 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∈ dom card)
17 simp3 1139 . . . . 5 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴𝑦𝐵 𝜑)
18 necom 2995 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∅ ≠ {𝑦𝐵𝜑})
19 rabn0 4384 . . . . . . . 8 ({𝑦𝐵𝜑} ≠ ∅ ↔ ∃𝑦𝐵 𝜑)
20 df-ne 2942 . . . . . . . 8 (∅ ≠ {𝑦𝐵𝜑} ↔ ¬ ∅ = {𝑦𝐵𝜑})
2118, 19, 203bitr3i 301 . . . . . . 7 (∃𝑦𝐵 𝜑 ↔ ¬ ∅ = {𝑦𝐵𝜑})
2221ralbii 3094 . . . . . 6 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑})
23 ralnex 3073 . . . . . 6 (∀𝑥𝐴 ¬ ∅ = {𝑦𝐵𝜑} ↔ ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2422, 23bitri 275 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2517, 24sylib 217 . . . 4 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ¬ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
26 0ex 5306 . . . . 5 ∅ ∈ V
2710elrnmpt 5953 . . . . 5 (∅ ∈ V → (∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ↔ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑}))
2826, 27ax-mp 5 . . . 4 (∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ↔ ∃𝑥𝐴 ∅ = {𝑦𝐵𝜑})
2925, 28sylnibr 329 . . 3 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ¬ ∅ ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}))
30 ac5num 10027 . . 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 585 . 2 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑔(𝑔:ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})⟶ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑧 ∈ ran (𝑥𝐴 ↦ {𝑦𝐵𝜑})(𝑔𝑧) ∈ 𝑧))
32 ffn 6714 . . . . . 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 1135 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∀𝑥𝐴 {𝑦𝐵𝜑} ∈ V)
35 fveq2 6888 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → (𝑔𝑧) = (𝑔‘{𝑦𝐵𝜑}))
36 id 22 . . . . . . . . 9 (𝑧 = {𝑦𝐵𝜑} → 𝑧 = {𝑦𝐵𝜑})
3735, 36eleq12d 2828 . . . . . . . 8 (𝑧 = {𝑦𝐵𝜑} → ((𝑔𝑧) ∈ 𝑧 ↔ (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}))
3810, 37ralrnmptw 7091 . . . . . . 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, 40imbitrid 243 . . . 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 1192 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → 𝐴𝑉)
4342mptexd 7221 . . . . . 6 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∈ V)
44 elrabi 3676 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
4544ralimi 3084 . . . . . . . . 9 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
4645ad2antll 728 . . . . . . . 8 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵)
47 eqid 2733 . . . . . . . . 9 (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
4847fmpt 7105 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵)
4946, 48sylib 217 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵)
50 nfcv 2904 . . . . . . . . . . 11 𝑦𝐵
5150elrabsf 3824 . . . . . . . . . 10 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} ↔ ((𝑔‘{𝑦𝐵𝜑}) ∈ 𝐵[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
5251simprbi 498 . . . . . . . . 9 ((𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
5352ralimi 3084 . . . . . . . 8 (∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑} → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
5453ad2antll 728 . . . . . . 7 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)
5549, 54jca 513 . . . . . 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))
56 feq1 6695 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓:𝐴𝐵 ↔ (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵))
57 nfmpt1 5255 . . . . . . . . 9 𝑥(𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
5857nfeq2 2921 . . . . . . . 8 𝑥 𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))
59 fvex 6901 . . . . . . . . . 10 (𝑓𝑥) ∈ V
60 ac6num.1 . . . . . . . . . 10 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
6159, 60sbcie 3819 . . . . . . . . 9 ([(𝑓𝑥) / 𝑦]𝜑𝜓)
62 fveq1 6887 . . . . . . . . . . 11 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (𝑓𝑥) = ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥))
63 fvex 6901 . . . . . . . . . . . 12 (𝑔‘{𝑦𝐵𝜑}) ∈ V
6447fvmpt2 7005 . . . . . . . . . . . 12 ((𝑥𝐴 ∧ (𝑔‘{𝑦𝐵𝜑}) ∈ V) → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
6563, 64mpan2 690 . . . . . . . . . . 11 (𝑥𝐴 → ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑}))‘𝑥) = (𝑔‘{𝑦𝐵𝜑}))
6662, 65sylan9eq 2793 . . . . . . . . . 10 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝑓𝑥) = (𝑔‘{𝑦𝐵𝜑}))
6766sbceq1d 3781 . . . . . . . . 9 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → ([(𝑓𝑥) / 𝑦]𝜑[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
6861, 67bitr3id 285 . . . . . . . 8 ((𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) ∧ 𝑥𝐴) → (𝜓[(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
6958, 68ralbida 3268 . . . . . . 7 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑))
7056, 69anbi12d 632 . . . . . 6 (𝑓 = (𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})) → ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓) ↔ ((𝑥𝐴 ↦ (𝑔‘{𝑦𝐵𝜑})):𝐴𝐵 ∧ ∀𝑥𝐴 [(𝑔‘{𝑦𝐵𝜑}) / 𝑦]𝜑)))
7143, 55, 70spcedv 3588 . . . . 5 (((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) ∧ (𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑})) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
7271ex 414 . . . 4 ((𝐴𝑉 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ((𝑔 Fn ran (𝑥𝐴 ↦ {𝑦𝐵𝜑}) ∧ ∀𝑥𝐴 (𝑔‘{𝑦𝐵𝜑}) ∈ {𝑦𝐵𝜑}) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))
7341, 72syld 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)))
7473exlimdv 1937 . 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)))
7531, 74mpd 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 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2941  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  [wsbc 3776  wss 3947  c0 4321   cuni 4907   ciun 4996  cmpt 5230  dom cdm 5675  ran crn 5676   Fn wfn 6535  wf 6536  cfv 6540  cardccrd 9926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-en 8936  df-card 9930
This theorem is referenced by:  ac6  10471  ptcmplem3  23540  poimirlem32  36458
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