Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > funopg | Unicode version |
Description: A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
funopg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3752 | . . . . 5 | |
2 | 1 | funeqd 5204 | . . . 4 |
3 | eqeq1 2171 | . . . 4 | |
4 | 2, 3 | imbi12d 233 | . . 3 |
5 | opeq2 3753 | . . . . 5 | |
6 | 5 | funeqd 5204 | . . . 4 |
7 | eqeq2 2174 | . . . 4 | |
8 | 6, 7 | imbi12d 233 | . . 3 |
9 | funrel 5199 | . . . . 5 | |
10 | vex 2724 | . . . . . 6 | |
11 | vex 2724 | . . . . . 6 | |
12 | 10, 11 | relop 4748 | . . . . 5 |
13 | 9, 12 | sylib 121 | . . . 4 |
14 | 10, 11 | opth 4209 | . . . . . . . 8 |
15 | vex 2724 | . . . . . . . . . . . 12 | |
16 | 15 | opid 3770 | . . . . . . . . . . 11 |
17 | 16 | preq1i 3650 | . . . . . . . . . 10 |
18 | vex 2724 | . . . . . . . . . . . 12 | |
19 | 15, 18 | dfop 3751 | . . . . . . . . . . 11 |
20 | 19 | preq2i 3651 | . . . . . . . . . 10 |
21 | 15 | snex 4158 | . . . . . . . . . . 11 |
22 | zfpair2 4182 | . . . . . . . . . . 11 | |
23 | 21, 22 | dfop 3751 | . . . . . . . . . 10 |
24 | 17, 20, 23 | 3eqtr4ri 2196 | . . . . . . . . 9 |
25 | 24 | eqeq2i 2175 | . . . . . . . 8 |
26 | 14, 25 | bitr3i 185 | . . . . . . 7 |
27 | dffun4 5193 | . . . . . . . . 9 | |
28 | 27 | simprbi 273 | . . . . . . . 8 |
29 | 15, 15 | opex 4201 | . . . . . . . . . . 11 |
30 | 29 | prid1 3676 | . . . . . . . . . 10 |
31 | eleq2 2228 | . . . . . . . . . 10 | |
32 | 30, 31 | mpbiri 167 | . . . . . . . . 9 |
33 | 15, 18 | opex 4201 | . . . . . . . . . . 11 |
34 | 33 | prid2 3677 | . . . . . . . . . 10 |
35 | eleq2 2228 | . . . . . . . . . 10 | |
36 | 34, 35 | mpbiri 167 | . . . . . . . . 9 |
37 | 32, 36 | jca 304 | . . . . . . . 8 |
38 | opeq12 3754 | . . . . . . . . . . . . . 14 | |
39 | 38 | 3adant3 1006 | . . . . . . . . . . . . 13 |
40 | 39 | eleq1d 2233 | . . . . . . . . . . . 12 |
41 | opeq12 3754 | . . . . . . . . . . . . . 14 | |
42 | 41 | 3adant2 1005 | . . . . . . . . . . . . 13 |
43 | 42 | eleq1d 2233 | . . . . . . . . . . . 12 |
44 | 40, 43 | anbi12d 465 | . . . . . . . . . . 11 |
45 | eqeq12 2177 | . . . . . . . . . . . 12 | |
46 | 45 | 3adant1 1004 | . . . . . . . . . . 11 |
47 | 44, 46 | imbi12d 233 | . . . . . . . . . 10 |
48 | 47 | spc3gv 2814 | . . . . . . . . 9 |
49 | 15, 15, 18, 48 | mp3an 1326 | . . . . . . . 8 |
50 | 28, 37, 49 | syl2im 38 | . . . . . . 7 |
51 | 26, 50 | syl5bi 151 | . . . . . 6 |
52 | dfsn2 3584 | . . . . . . . . . . 11 | |
53 | preq2 3648 | . . . . . . . . . . 11 | |
54 | 52, 53 | eqtr2id 2210 | . . . . . . . . . 10 |
55 | 54 | eqeq2d 2176 | . . . . . . . . 9 |
56 | eqtr3 2184 | . . . . . . . . . 10 | |
57 | 56 | expcom 115 | . . . . . . . . 9 |
58 | 55, 57 | syl6bi 162 | . . . . . . . 8 |
59 | 58 | com13 80 | . . . . . . 7 |
60 | 59 | imp 123 | . . . . . 6 |
61 | 51, 60 | sylcom 28 | . . . . 5 |
62 | 61 | exlimdvv 1884 | . . . 4 |
63 | 13, 62 | mpd 13 | . . 3 |
64 | 4, 8, 63 | vtocl2g 2785 | . 2 |
65 | 64 | 3impia 1189 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wal 1340 wceq 1342 wex 1479 wcel 2135 cvv 2721 csn 3570 cpr 3571 cop 3573 wrel 4603 wfun 5176 |
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-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
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-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-fun 5184 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |