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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 5460 | . . . . . 6 | |
2 | df-mpt 3986 | . . . . . 6 | |
3 | 1, 2 | syl6eq 2186 | . . . . 5 |
4 | dffn5im 5460 | . . . . . 6 | |
5 | df-mpt 3986 | . . . . . 6 | |
6 | 4, 5 | syl6eq 2186 | . . . . 5 |
7 | 3, 6 | ineqan12d 3274 | . . . 4 |
8 | inopab 4666 | . . . 4 | |
9 | 7, 8 | syl6eq 2186 | . . 3 |
10 | 9 | dmeqd 4736 | . 2 |
11 | anandi 579 | . . . . . . . 8 | |
12 | 11 | exbii 1584 | . . . . . . 7 |
13 | 19.42v 1878 | . . . . . . 7 | |
14 | 12, 13 | bitr3i 185 | . . . . . 6 |
15 | funfvex 5431 | . . . . . . . . 9 | |
16 | eqeq1 2144 | . . . . . . . . . 10 | |
17 | 16 | ceqsexgv 2809 | . . . . . . . . 9 |
18 | 15, 17 | syl 14 | . . . . . . . 8 |
19 | 18 | funfni 5218 | . . . . . . 7 |
20 | 19 | pm5.32da 447 | . . . . . 6 |
21 | 14, 20 | syl5bb 191 | . . . . 5 |
22 | 21 | abbidv 2255 | . . . 4 |
23 | dmopab 4745 | . . . 4 | |
24 | df-rab 2423 | . . . 4 | |
25 | 22, 23, 24 | 3eqtr4g 2195 | . . 3 |
26 | 25 | adantr 274 | . 2 |
27 | 10, 26 | eqtrd 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cab 2123 crab 2418 cvv 2681 cin 3065 copab 3983 cmpt 3984 cdm 4534 wfun 5112 wfn 5113 cfv 5118 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 |
This theorem is referenced by: fneqeql 5521 |
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