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 | ⊢ ( 𝑅 ∈ 𝐴 → ∃ 𝑓 ( 𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 | ⊢ ( 𝑥 = 𝑅 → ( 𝑓 ⊆ 𝑥 ↔ 𝑓 ⊆ 𝑅 ) ) | |
2 | dmeq | ⊢ ( 𝑥 = 𝑅 → dom 𝑥 = dom 𝑅 ) | |
3 | 2 | fneq2d | ⊢ ( 𝑥 = 𝑅 → ( 𝑓 Fn dom 𝑥 ↔ 𝑓 Fn dom 𝑅 ) ) |
4 | 1 3 | anbi12d | ⊢ ( 𝑥 = 𝑅 → ( ( 𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥 ) ↔ ( 𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅 ) ) ) |
5 | 4 | exbidv | ⊢ ( 𝑥 = 𝑅 → ( ∃ 𝑓 ( 𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥 ) ↔ ∃ 𝑓 ( 𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅 ) ) ) |
6 | ac7 | ⊢ ∃ 𝑓 ( 𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥 ) | |
7 | 5 6 | vtoclg | ⊢ ( 𝑅 ∈ 𝐴 → ∃ 𝑓 ( 𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅 ) ) |