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Theorem ac7g 10407
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 3969 . . . 4 (𝑥 = 𝑅 → (𝑓𝑥𝑓𝑅))
2 dmeq 5858 . . . . 5 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
32fneq2d 6594 . . . 4 (𝑥 = 𝑅 → (𝑓 Fn dom 𝑥𝑓 Fn dom 𝑅))
41, 3anbi12d 631 . . 3 (𝑥 = 𝑅 → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓𝑅𝑓 Fn dom 𝑅)))
54exbidv 1924 . 2 (𝑥 = 𝑅 → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅)))
6 ac7 10406 . 2 𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
75, 6vtoclg 3524 1 (𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  wss 3909  dom cdm 5632   Fn wfn 6489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669  ax-ac2 10396
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-ac 10049
This theorem is referenced by: (None)
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