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Theorem ac6 10094
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 10098, 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 3993 . . . . . 6 {𝑦𝐵𝜑} ⊆ 𝐵
43rgenw 3073 . . . . 5 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵
5 iunss 4954 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵)
64, 5mpbir 234 . . . 4 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵
72, 6ssexi 5215 . . 3 𝑥𝐴 {𝑦𝐵𝜑} ∈ V
8 numth3 10084 . . 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 10093 . 2 ((𝐴 ∈ V ∧ 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
121, 9, 11mp3an12 1453 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2110  wral 3061  wrex 3062  {crab 3065  Vcvv 3408  wss 3866   ciun 4904  dom cdm 5551  wf 6376  cfv 6380  cardccrd 9551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-ac2 10077
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-wrecs 8047  df-recs 8108  df-en 8627  df-card 9555  df-ac 9730
This theorem is referenced by:  ac6c4  10095  ac6s  10098  wlkiswwlksupgr2  27961
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