Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ac7g | Structured version Visualization version GIF version |
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 |
---|---|
ac7g | ⊢ (𝑅 ∈ 𝐴 → ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3992 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝑓 ⊆ 𝑥 ↔ 𝑓 ⊆ 𝑅)) | |
2 | dmeq 5766 | . . . . 5 ⊢ (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅) | |
3 | 2 | fneq2d 6441 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝑓 Fn dom 𝑥 ↔ 𝑓 Fn dom 𝑅)) |
4 | 1, 3 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑅 → ((𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) ↔ (𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅))) |
5 | 4 | exbidv 1918 | . 2 ⊢ (𝑥 = 𝑅 → (∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) ↔ ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅))) |
6 | ac7 9889 | . 2 ⊢ ∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥) | |
7 | 5, 6 | vtoclg 3567 | 1 ⊢ (𝑅 ∈ 𝐴 → ∃𝑓(𝑓 ⊆ 𝑅 ∧ 𝑓 Fn dom 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ⊆ wss 3935 dom cdm 5549 Fn wfn 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-ac2 9879 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ac 9536 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |