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Mirrors > Home > ILE Home > Th. List > acfun | Unicode version |
Description: A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.) |
Ref | Expression |
---|---|
acfun.ac | CHOICE |
acfun.a | |
acfun.m |
Ref | Expression |
---|---|
acfun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acfun.a | . . . . 5 | |
2 | 1 | elexd 2748 | . . . 4 |
3 | abid2 2296 | . . . . . 6 | |
4 | vex 2738 | . . . . . 6 | |
5 | 3, 4 | eqeltri 2248 | . . . . 5 |
6 | 5 | a1i 9 | . . . 4 |
7 | 2, 6 | opabex3d 6112 | . . 3 |
8 | acfun.ac | . . . 4 CHOICE | |
9 | df-ac 7195 | . . . 4 CHOICE | |
10 | 8, 9 | sylib 122 | . . 3 |
11 | sseq2 3177 | . . . . . 6 | |
12 | dmeq 4820 | . . . . . . 7 | |
13 | 12 | fneq2d 5299 | . . . . . 6 |
14 | 11, 13 | anbi12d 473 | . . . . 5 |
15 | 14 | exbidv 1823 | . . . 4 |
16 | 15 | spcgv 2822 | . . 3 |
17 | 7, 10, 16 | sylc 62 | . 2 |
18 | simprr 531 | . . . . . 6 | |
19 | acfun.m | . . . . . . . . . 10 | |
20 | elequ2 2151 | . . . . . . . . . . . . 13 | |
21 | 20 | exbidv 1823 | . . . . . . . . . . . 12 |
22 | 21 | cbvralv 2701 | . . . . . . . . . . 11 |
23 | elequ1 2150 | . . . . . . . . . . . . 13 | |
24 | 23 | cbvexv 1916 | . . . . . . . . . . . 12 |
25 | 24 | ralbii 2481 | . . . . . . . . . . 11 |
26 | 22, 25 | bitri 184 | . . . . . . . . . 10 |
27 | 19, 26 | sylib 122 | . . . . . . . . 9 |
28 | dmopab3 4833 | . . . . . . . . 9 | |
29 | 27, 28 | sylib 122 | . . . . . . . 8 |
30 | 29 | fneq2d 5299 | . . . . . . 7 |
31 | 30 | adantr 276 | . . . . . 6 |
32 | 18, 31 | mpbid 147 | . . . . 5 |
33 | simplrl 535 | . . . . . . . . 9 | |
34 | fnopfv 5638 | . . . . . . . . . 10 | |
35 | 32, 34 | sylan 283 | . . . . . . . . 9 |
36 | 33, 35 | sseldd 3154 | . . . . . . . 8 |
37 | vex 2738 | . . . . . . . . 9 | |
38 | vex 2738 | . . . . . . . . . 10 | |
39 | 38, 37 | fvex 5527 | . . . . . . . . 9 |
40 | eleq1 2238 | . . . . . . . . . 10 | |
41 | elequ2 2151 | . . . . . . . . . 10 | |
42 | 40, 41 | anbi12d 473 | . . . . . . . . 9 |
43 | eleq1 2238 | . . . . . . . . . 10 | |
44 | 43 | anbi2d 464 | . . . . . . . . 9 |
45 | 37, 39, 42, 44 | opelopab 4265 | . . . . . . . 8 |
46 | 36, 45 | sylib 122 | . . . . . . 7 |
47 | 46 | simprd 114 | . . . . . 6 |
48 | 47 | ralrimiva 2548 | . . . . 5 |
49 | 32, 48 | jca 306 | . . . 4 |
50 | 49 | ex 115 | . . 3 |
51 | 50 | eximdv 1878 | . 2 |
52 | 17, 51 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wal 1351 wceq 1353 wex 1490 wcel 2146 cab 2161 wral 2453 cvv 2735 wss 3127 cop 3592 copab 4058 cdm 4620 wfn 5203 cfv 5208 CHOICEwac 7194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-ac 7195 |
This theorem is referenced by: exmidaclem 7197 |
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