Theorem ac7g 10387

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 3949	. . . 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 633	. . 3 ⊢ (𝑥 = 𝑅 → ((𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) ↔ (𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅)))
5	4	exbidv 1923	. 2 ⊢ (𝑥 = 𝑅 → (∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅)))
6		ac7 10386	. 2 ⊢ ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
7	5, 6	vtoclg 3500	1 ⊢ (𝑅 ∈ 𝐴 → ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ⊆ wss 3890  dom cdm 5624   Fn wfn 6487

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ac 10029

This theorem is referenced by: (None)