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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 3202 | . . . . 5 | |
2 | dmss 4738 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | fndm 5222 | . . . . 5 | |
5 | 4 | adantr 274 | . . . 4 |
6 | 3, 5 | sseqtrid 3147 | . . 3 |
7 | dfss1 3280 | . . 3 | |
8 | 6, 7 | sylib 121 | . 2 |
9 | vex 2689 | . . . . 5 | |
10 | 9 | eldm 4736 | . . . 4 |
11 | eqcom 2141 | . . . . . . . 8 | |
12 | fnbrfvb 5462 | . . . . . . . 8 | |
13 | 11, 12 | syl5bb 191 | . . . . . . 7 |
14 | 13 | adantll 467 | . . . . . 6 |
15 | 14 | necon3abid 2347 | . . . . 5 |
16 | funfvex 5438 | . . . . . . . 8 | |
17 | 16 | funfni 5223 | . . . . . . 7 |
18 | 17 | adantlr 468 | . . . . . 6 |
19 | breq2 3933 | . . . . . . . 8 | |
20 | 19 | notbid 656 | . . . . . . 7 |
21 | 20 | ceqsexgv 2814 | . . . . . 6 |
22 | 18, 21 | syl 14 | . . . . 5 |
23 | eqcom 2141 | . . . . . . . . . 10 | |
24 | fnbrfvb 5462 | . . . . . . . . . 10 | |
25 | 23, 24 | syl5bb 191 | . . . . . . . . 9 |
26 | 25 | adantlr 468 | . . . . . . . 8 |
27 | 26 | anbi1d 460 | . . . . . . 7 |
28 | brdif 3981 | . . . . . . 7 | |
29 | 27, 28 | syl6bbr 197 | . . . . . 6 |
30 | 29 | exbidv 1797 | . . . . 5 |
31 | 15, 22, 30 | 3bitr2rd 216 | . . . 4 |
32 | 10, 31 | syl5bb 191 | . . 3 |
33 | 32 | rabbi2dva 3284 | . 2 |
34 | 8, 33 | eqtr3d 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wne 2308 crab 2420 cvv 2686 cdif 3068 cin 3070 wss 3071 class class class wbr 3929 cdm 4539 wfn 5118 cfv 5123 |
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-in1 603 ax-in2 604 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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: fndmdifcom 5526 |
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