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Theorem ac6 9492
Description: Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set 𝐵, where 𝜑 depends on 𝑥 (the natural number) and 𝑦 (to specify a member of 𝐵). A stronger version of this theorem, ac6s 9496, allows 𝐵 to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
ac6.1 𝐴 ∈ V
ac6.2 𝐵 ∈ V
ac6.3 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
Assertion
Ref Expression
ac6 (∀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)

Proof of Theorem ac6
StepHypRef Expression
1 ac6.1 . 2 𝐴 ∈ V
2 ac6.2 . . . 4 𝐵 ∈ V
3 ssrab2 3826 . . . . . 6 {𝑦𝐵𝜑} ⊆ 𝐵
43rgenw 3060 . . . . 5 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵
5 iunss 4711 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵)
64, 5mpbir 221 . . . 4 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵
72, 6ssexi 4953 . . 3 𝑥𝐴 {𝑦𝐵𝜑} ∈ V
8 numth3 9482 . . 3 ( 𝑥𝐴 {𝑦𝐵𝜑} ∈ V → 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card)
97, 8ax-mp 5 . 2 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card
10 ac6.3 . . 3 (𝑦 = (𝑓𝑥) → (𝜑𝜓))
1110ac6num 9491 . 2 ((𝐴 ∈ V ∧ 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
121, 9, 11mp3an12 1561 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wex 1851  wcel 2137  wral 3048  wrex 3049  {crab 3052  Vcvv 3338  wss 3713   ciun 4670  dom cdm 5264  wf 6043  cfv 6047  cardccrd 8949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-ac2 9475
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-wrecs 7574  df-recs 7635  df-en 8120  df-card 8953  df-ac 9127
This theorem is referenced by:  ac6c4  9493  ac6s  9496  wlkiswwlksupgr2  26984
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