Theorem ac7g 10394

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	⊢ (𝑅 ∈ 𝐴 → ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 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.

Step	Hyp	Ref	Expression
1		sseq2 3948	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑓 ⊆ 𝑥 ↔ 𝑓 ⊆ 𝑅))
2		dmeq 5852	. . . . 5 ⊢ (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3	2	fneq2d 6586	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑓 Fn dom 𝑥 ↔ 𝑓 Fn dom 𝑅))
4	1, 3	anbi12d 638	. . 3 ⊢ (𝑥 = 𝑅 → ((𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) ↔ (𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅)))
5	4	exbidv 1928	. 2 ⊢ (𝑥 = 𝑅 → (∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅)))
6		ac7 10393	. 2 ⊢ ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
7	5, 6	vtoclg 3502	1 ⊢ (𝑅 ∈ 𝐴 → ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ⊆ wss 3890  dom cdm 5625   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-ac2 10383

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ac 10036

This theorem is referenced by: (None)