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Theorem ac7g 9884
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 3990 . . . 4 (𝑥 = 𝑅 → (𝑓𝑥𝑓𝑅))
2 dmeq 5765 . . . . 5 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
32fneq2d 6440 . . . 4 (𝑥 = 𝑅 → (𝑓 Fn dom 𝑥𝑓 Fn dom 𝑅))
41, 3anbi12d 630 . . 3 (𝑥 = 𝑅 → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓𝑅𝑓 Fn dom 𝑅)))
54exbidv 1913 . 2 (𝑥 = 𝑅 → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅)))
6 ac7 9883 . 2 𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
75, 6vtoclg 3565 1 (𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wss 3933  dom cdm 5548   Fn wfn 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ac 9530
This theorem is referenced by: (None)
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