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Mirrors > Home > ILE Home > Th. List > fndmin | Unicode version |
Description: Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
fndmin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn5im 5435 | . . . . . 6 | |
2 | df-mpt 3961 | . . . . . 6 | |
3 | 1, 2 | syl6eq 2166 | . . . . 5 |
4 | dffn5im 5435 | . . . . . 6 | |
5 | df-mpt 3961 | . . . . . 6 | |
6 | 4, 5 | syl6eq 2166 | . . . . 5 |
7 | 3, 6 | ineqan12d 3249 | . . . 4 |
8 | inopab 4641 | . . . 4 | |
9 | 7, 8 | syl6eq 2166 | . . 3 |
10 | 9 | dmeqd 4711 | . 2 |
11 | anandi 564 | . . . . . . . 8 | |
12 | 11 | exbii 1569 | . . . . . . 7 |
13 | 19.42v 1862 | . . . . . . 7 | |
14 | 12, 13 | bitr3i 185 | . . . . . 6 |
15 | funfvex 5406 | . . . . . . . . 9 | |
16 | eqeq1 2124 | . . . . . . . . . 10 | |
17 | 16 | ceqsexgv 2788 | . . . . . . . . 9 |
18 | 15, 17 | syl 14 | . . . . . . . 8 |
19 | 18 | funfni 5193 | . . . . . . 7 |
20 | 19 | pm5.32da 447 | . . . . . 6 |
21 | 14, 20 | syl5bb 191 | . . . . 5 |
22 | 21 | abbidv 2235 | . . . 4 |
23 | dmopab 4720 | . . . 4 | |
24 | df-rab 2402 | . . . 4 | |
25 | 22, 23, 24 | 3eqtr4g 2175 | . . 3 |
26 | 25 | adantr 274 | . 2 |
27 | 10, 26 | eqtrd 2150 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cab 2103 crab 2397 cvv 2660 cin 3040 copab 3958 cmpt 3959 cdm 4509 wfun 5087 wfn 5088 cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 |
This theorem is referenced by: fneqeql 5496 |
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