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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 2734 | . . . 4 |
3 | abid2 2285 | . . . . . 6 | |
4 | vex 2724 | . . . . . 6 | |
5 | 3, 4 | eqeltri 2237 | . . . . 5 |
6 | 5 | a1i 9 | . . . 4 |
7 | 2, 6 | opabex3d 6081 | . . 3 |
8 | acfun.ac | . . . 4 CHOICE | |
9 | df-ac 7153 | . . . 4 CHOICE | |
10 | 8, 9 | sylib 121 | . . 3 |
11 | sseq2 3161 | . . . . . 6 | |
12 | dmeq 4798 | . . . . . . 7 | |
13 | 12 | fneq2d 5273 | . . . . . 6 |
14 | 11, 13 | anbi12d 465 | . . . . 5 |
15 | 14 | exbidv 1812 | . . . 4 |
16 | 15 | spcgv 2808 | . . 3 |
17 | 7, 10, 16 | sylc 62 | . 2 |
18 | simprr 522 | . . . . . 6 | |
19 | acfun.m | . . . . . . . . . 10 | |
20 | elequ2 2140 | . . . . . . . . . . . . 13 | |
21 | 20 | exbidv 1812 | . . . . . . . . . . . 12 |
22 | 21 | cbvralv 2689 | . . . . . . . . . . 11 |
23 | elequ1 2139 | . . . . . . . . . . . . 13 | |
24 | 23 | cbvexv 1905 | . . . . . . . . . . . 12 |
25 | 24 | ralbii 2470 | . . . . . . . . . . 11 |
26 | 22, 25 | bitri 183 | . . . . . . . . . 10 |
27 | 19, 26 | sylib 121 | . . . . . . . . 9 |
28 | dmopab3 4811 | . . . . . . . . 9 | |
29 | 27, 28 | sylib 121 | . . . . . . . 8 |
30 | 29 | fneq2d 5273 | . . . . . . 7 |
31 | 30 | adantr 274 | . . . . . 6 |
32 | 18, 31 | mpbid 146 | . . . . 5 |
33 | simplrl 525 | . . . . . . . . 9 | |
34 | fnopfv 5609 | . . . . . . . . . 10 | |
35 | 32, 34 | sylan 281 | . . . . . . . . 9 |
36 | 33, 35 | sseldd 3138 | . . . . . . . 8 |
37 | vex 2724 | . . . . . . . . 9 | |
38 | vex 2724 | . . . . . . . . . 10 | |
39 | 38, 37 | fvex 5500 | . . . . . . . . 9 |
40 | eleq1 2227 | . . . . . . . . . 10 | |
41 | elequ2 2140 | . . . . . . . . . 10 | |
42 | 40, 41 | anbi12d 465 | . . . . . . . . 9 |
43 | eleq1 2227 | . . . . . . . . . 10 | |
44 | 43 | anbi2d 460 | . . . . . . . . 9 |
45 | 37, 39, 42, 44 | opelopab 4243 | . . . . . . . 8 |
46 | 36, 45 | sylib 121 | . . . . . . 7 |
47 | 46 | simprd 113 | . . . . . 6 |
48 | 47 | ralrimiva 2537 | . . . . 5 |
49 | 32, 48 | jca 304 | . . . 4 |
50 | 49 | ex 114 | . . 3 |
51 | 50 | eximdv 1867 | . 2 |
52 | 17, 51 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1340 wceq 1342 wex 1479 wcel 2135 cab 2150 wral 2442 cvv 2721 wss 3111 cop 3573 copab 4036 cdm 4598 wfn 5177 cfv 5182 CHOICEwac 7152 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ac 7153 |
This theorem is referenced by: exmidaclem 7155 |
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