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Theorem ac6 10521
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 10525, 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 4079 . . . . . 6 {𝑦𝐵𝜑} ⊆ 𝐵
43rgenw 3064 . . . . 5 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵
5 iunss 5044 . . . . 5 ( 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵)
64, 5mpbir 231 . . . 4 𝑥𝐴 {𝑦𝐵𝜑} ⊆ 𝐵
72, 6ssexi 5321 . . 3 𝑥𝐴 {𝑦𝐵𝜑} ∈ V
8 numth3 10511 . . 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 10520 . 2 ((𝐴 ∈ V ∧ 𝑥𝐴 {𝑦𝐵𝜑} ∈ dom card ∧ ∀𝑥𝐴𝑦𝐵 𝜑) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
121, 9, 11mp3an12 1452 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  wss 3950   ciun 4990  dom cdm 5684  wf 6556  cfv 6560  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-ac2 10504
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-en 8987  df-card 9980  df-ac 10157
This theorem is referenced by:  ac6c4  10522  ac6s  10525  wlkiswwlksupgr2  29898
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