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Theorem ac6 10518
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 10522, 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 4090 . . . . . 6 {𝑦𝐵𝜑} ⊆ 𝐵
43rgenw 3063 . . . . 5 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵
5 iunss 5050 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵)
64, 5mpbir 231 . . . 4 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵
72, 6ssexi 5328 . . 3 𝑥𝐴 {𝑦𝐵𝜑} ∈ V
8 numth3 10508 . . 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 10517 . 2 ((𝐴 ∈ V ∧ 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
121, 9, 11mp3an12 1450 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963   ciun 4996  dom cdm 5689  wf 6559  cfv 6563  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-en 8985  df-card 9977  df-ac 10154
This theorem is referenced by:  ac6c4  10519  ac6s  10522  wlkiswwlksupgr2  29907
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