MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac7g Structured version   Visualization version   GIF version

Theorem ac7g 9873
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 5745 . . . . 5 (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
32fneq2d 6420 . . . 4 (𝑥 = 𝑅 → (𝑓 Fn dom 𝑥𝑓 Fn dom 𝑅))
41, 3anbi12d 633 . . 3 (𝑥 = 𝑅 → ((𝑓𝑥𝑓 Fn dom 𝑥) ↔ (𝑓𝑅𝑓 Fn dom 𝑅)))
54exbidv 1923 . 2 (𝑥 = 𝑅 → (∃𝑓(𝑓𝑥𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅)))
6 ac7 9872 . 2 𝑓(𝑓𝑥𝑓 Fn dom 𝑥)
75, 6vtoclg 3544 1 (𝑅𝐴 → ∃𝑓(𝑓𝑅𝑓 Fn dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2115  wss 3910  dom cdm 5528   Fn wfn 6323
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-ac2 9862
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ac 9519
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator