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Mirrors > Home > ILE Home > Th. List > fndmdif | Unicode version |
Description: Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
fndmdif |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3070 | . . . . 5 | |
2 | dmss 4534 | . . . . 5 | |
3 | 1, 2 | ax-mp 7 | . . . 4 |
4 | fndm 4998 | . . . . 5 | |
5 | 4 | adantr 261 | . . . 4 |
6 | 3, 5 | syl5sseq 2993 | . . 3 |
7 | dfss1 3141 | . . 3 | |
8 | 6, 7 | sylib 127 | . 2 |
9 | vex 2560 | . . . . 5 | |
10 | 9 | eldm 4532 | . . . 4 |
11 | eqcom 2042 | . . . . . . . 8 | |
12 | fnbrfvb 5214 | . . . . . . . 8 | |
13 | 11, 12 | syl5bb 181 | . . . . . . 7 |
14 | 13 | adantll 445 | . . . . . 6 |
15 | 14 | necon3abid 2244 | . . . . 5 |
16 | funfvex 5192 | . . . . . . . 8 | |
17 | 16 | funfni 4999 | . . . . . . 7 |
18 | 17 | adantlr 446 | . . . . . 6 |
19 | breq2 3768 | . . . . . . . 8 | |
20 | 19 | notbid 592 | . . . . . . 7 |
21 | 20 | ceqsexgv 2673 | . . . . . 6 |
22 | 18, 21 | syl 14 | . . . . 5 |
23 | eqcom 2042 | . . . . . . . . . 10 | |
24 | fnbrfvb 5214 | . . . . . . . . . 10 | |
25 | 23, 24 | syl5bb 181 | . . . . . . . . 9 |
26 | 25 | adantlr 446 | . . . . . . . 8 |
27 | 26 | anbi1d 438 | . . . . . . 7 |
28 | brdif 3812 | . . . . . . 7 | |
29 | 27, 28 | syl6bbr 187 | . . . . . 6 |
30 | 29 | exbidv 1706 | . . . . 5 |
31 | 15, 22, 30 | 3bitr2rd 206 | . . . 4 |
32 | 10, 31 | syl5bb 181 | . . 3 |
33 | 32 | rabbi2dva 3145 | . 2 |
34 | 8, 33 | eqtr3d 2074 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 wne 2204 crab 2310 cvv 2557 cdif 2914 cin 2916 wss 2917 class class class wbr 3764 cdm 4345 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-in1 544 ax-in2 545 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-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 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-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: fndmdifcom 5273 |
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