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Theorem ac7g 10374
Description: An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
ac7g (𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))
Distinct variable group:   𝑅,𝑓
Allowed substitution hint:   𝐴(𝑓)

Proof of Theorem ac7g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sseq2 3957 . . . 4 (𝑥 = 𝑅 → (𝑓𝑥𝑓𝑅))
2 dmeq 5849 . . . . 5 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
32fneq2d 6582 . . . 4 (𝑥 = 𝑅 → (𝑓 Fn dom 𝑥𝑓 Fn dom 𝑅))
41, 3anbi12d 632 . . 3 (𝑥 = 𝑅 → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓𝑅𝑓 Fn dom 𝑅)))
54exbidv 1922 . 2 (𝑥 = 𝑅 → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅)))
6 ac7 10373 . 2 𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
75, 6vtoclg 3508 1 (𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1541  wex 1780  wcel 2113  wss 3898  dom cdm 5621   Fn wfn 6483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-ac2 10363
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-ac 10016
This theorem is referenced by: (None)
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