Metamath Proof Explorer


Theorem ac7g

Description: An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of Enderton p. 49. (Contributed by NM, 23-Jul-2004)

Ref Expression
Assertion ac7g ( 𝑅𝐴 → ∃ 𝑓 ( 𝑓𝑅𝑓 Fn dom 𝑅 ) )

Proof

Step Hyp Ref Expression
1 sseq2 ( 𝑥 = 𝑅 → ( 𝑓𝑥𝑓𝑅 ) )
2 dmeq ( 𝑥 = 𝑅 → dom 𝑥 = dom 𝑅 )
3 2 fneq2d ( 𝑥 = 𝑅 → ( 𝑓 Fn dom 𝑥𝑓 Fn dom 𝑅 ) )
4 1 3 anbi12d ( 𝑥 = 𝑅 → ( ( 𝑓𝑥𝑓 Fn dom 𝑥 ) ↔ ( 𝑓𝑅𝑓 Fn dom 𝑅 ) ) )
5 4 exbidv ( 𝑥 = 𝑅 → ( ∃ 𝑓 ( 𝑓𝑥𝑓 Fn dom 𝑥 ) ↔ ∃ 𝑓 ( 𝑓𝑅𝑓 Fn dom 𝑅 ) ) )
6 ac7 𝑓 ( 𝑓𝑥𝑓 Fn dom 𝑥 )
7 5 6 vtoclg ( 𝑅𝐴 → ∃ 𝑓 ( 𝑓𝑅𝑓 Fn dom 𝑅 ) )