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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 2699 | . . . 4 |
3 | abid2 2260 | . . . . . 6 | |
4 | vex 2689 | . . . . . 6 | |
5 | 3, 4 | eqeltri 2212 | . . . . 5 |
6 | 5 | a1i 9 | . . . 4 |
7 | 2, 6 | opabex3d 6019 | . . 3 |
8 | acfun.ac | . . . 4 CHOICE | |
9 | df-ac 7062 | . . . 4 CHOICE | |
10 | 8, 9 | sylib 121 | . . 3 |
11 | sseq2 3121 | . . . . . 6 | |
12 | dmeq 4739 | . . . . . . 7 | |
13 | 12 | fneq2d 5214 | . . . . . 6 |
14 | 11, 13 | anbi12d 464 | . . . . 5 |
15 | 14 | exbidv 1797 | . . . 4 |
16 | 15 | spcgv 2773 | . . 3 |
17 | 7, 10, 16 | sylc 62 | . 2 |
18 | simprr 521 | . . . . . 6 | |
19 | acfun.m | . . . . . . . . . 10 | |
20 | elequ2 1691 | . . . . . . . . . . . . 13 | |
21 | 20 | exbidv 1797 | . . . . . . . . . . . 12 |
22 | 21 | cbvralv 2654 | . . . . . . . . . . 11 |
23 | elequ1 1690 | . . . . . . . . . . . . 13 | |
24 | 23 | cbvexv 1890 | . . . . . . . . . . . 12 |
25 | 24 | ralbii 2441 | . . . . . . . . . . 11 |
26 | 22, 25 | bitri 183 | . . . . . . . . . 10 |
27 | 19, 26 | sylib 121 | . . . . . . . . 9 |
28 | dmopab3 4752 | . . . . . . . . 9 | |
29 | 27, 28 | sylib 121 | . . . . . . . 8 |
30 | 29 | fneq2d 5214 | . . . . . . 7 |
31 | 30 | adantr 274 | . . . . . 6 |
32 | 18, 31 | mpbid 146 | . . . . 5 |
33 | simplrl 524 | . . . . . . . . 9 | |
34 | fnopfv 5550 | . . . . . . . . . 10 | |
35 | 32, 34 | sylan 281 | . . . . . . . . 9 |
36 | 33, 35 | sseldd 3098 | . . . . . . . 8 |
37 | vex 2689 | . . . . . . . . 9 | |
38 | vex 2689 | . . . . . . . . . 10 | |
39 | 38, 37 | fvex 5441 | . . . . . . . . 9 |
40 | eleq1 2202 | . . . . . . . . . 10 | |
41 | elequ2 1691 | . . . . . . . . . 10 | |
42 | 40, 41 | anbi12d 464 | . . . . . . . . 9 |
43 | eleq1 2202 | . . . . . . . . . 10 | |
44 | 43 | anbi2d 459 | . . . . . . . . 9 |
45 | 37, 39, 42, 44 | opelopab 4193 | . . . . . . . 8 |
46 | 36, 45 | sylib 121 | . . . . . . 7 |
47 | 46 | simprd 113 | . . . . . 6 |
48 | 47 | ralrimiva 2505 | . . . . 5 |
49 | 32, 48 | jca 304 | . . . 4 |
50 | 49 | ex 114 | . . 3 |
51 | 50 | eximdv 1852 | . 2 |
52 | 17, 51 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wex 1468 wcel 1480 cab 2125 wral 2416 cvv 2686 wss 3071 cop 3530 copab 3988 cdm 4539 wfn 5118 cfv 5123 CHOICEwac 7061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ac 7062 |
This theorem is referenced by: exmidaclem 7064 |
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