Theorem ac7g 10469

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 3964	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑓 ⊆ 𝑥 ↔ 𝑓 ⊆ 𝑅))
2		dmeq 5895	. . . . 5 ⊢ (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅)
3	2	fneq2d 6633	. . . 4 ⊢ (𝑥 = 𝑅 → (𝑓 Fn dom 𝑥 ↔ 𝑓 Fn dom 𝑅))
4	1, 3	anbi12d 644	. . 3 ⊢ (𝑥 = 𝑅 → ((𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) ↔ (𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅)))
5	4	exbidv 1954	. 2 ⊢ (𝑥 = 𝑅 → (∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅)))
6		ac7 10468	. 2 ⊢ ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)
7	5, 6	vtoclg 3524	1 ⊢ (𝑅 ∈ 𝐴 → ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ⊆ wss 3906  dom cdm 5663   Fn wfn 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-ac2 10458

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ac 10112

This theorem is used by: (None)