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Theorem ac7g 10272
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 3952 . . . 4 (𝑥 = 𝑅 → (𝑓𝑥𝑓𝑅))
2 dmeq 5821 . . . . 5 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
32fneq2d 6554 . . . 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 10271 . 2 𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
75, 6vtoclg 3510 1 (𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1539  wex 1779  wcel 2104  wss 3892  dom cdm 5596   Fn wfn 6449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616  ax-ac2 10261
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5496  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-fv 6462  df-ac 9914
This theorem is referenced by: (None)
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