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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 5219 | . . . . . 6 | |
2 | df-mpt 3820 | . . . . . 6 | |
3 | 1, 2 | syl6eq 2088 | . . . . 5 |
4 | dffn5im 5219 | . . . . . 6 | |
5 | df-mpt 3820 | . . . . . 6 | |
6 | 4, 5 | syl6eq 2088 | . . . . 5 |
7 | 3, 6 | ineqan12d 3140 | . . . 4 |
8 | inopab 4468 | . . . 4 | |
9 | 7, 8 | syl6eq 2088 | . . 3 |
10 | 9 | dmeqd 4537 | . 2 |
11 | anandi 524 | . . . . . . . 8 | |
12 | 11 | exbii 1496 | . . . . . . 7 |
13 | 19.42v 1786 | . . . . . . 7 | |
14 | 12, 13 | bitr3i 175 | . . . . . 6 |
15 | funfvex 5192 | . . . . . . . . 9 | |
16 | eqeq1 2046 | . . . . . . . . . 10 | |
17 | 16 | ceqsexgv 2673 | . . . . . . . . 9 |
18 | 15, 17 | syl 14 | . . . . . . . 8 |
19 | 18 | funfni 4999 | . . . . . . 7 |
20 | 19 | pm5.32da 425 | . . . . . 6 |
21 | 14, 20 | syl5bb 181 | . . . . 5 |
22 | 21 | abbidv 2155 | . . . 4 |
23 | dmopab 4546 | . . . 4 | |
24 | df-rab 2315 | . . . 4 | |
25 | 22, 23, 24 | 3eqtr4g 2097 | . . 3 |
26 | 25 | adantr 261 | . 2 |
27 | 10, 26 | eqtrd 2072 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cab 2026 crab 2310 cvv 2557 cin 2916 copab 3817 cmpt 3818 cdm 4345 wfun 4896 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: fneqeql 5275 |
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