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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 3234 | . . . . 5 | |
2 | dmss 4788 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | fndm 5272 | . . . . 5 | |
5 | 4 | adantr 274 | . . . 4 |
6 | 3, 5 | sseqtrid 3178 | . . 3 |
7 | dfss1 3312 | . . 3 | |
8 | 6, 7 | sylib 121 | . 2 |
9 | vex 2715 | . . . . 5 | |
10 | 9 | eldm 4786 | . . . 4 |
11 | eqcom 2159 | . . . . . . . 8 | |
12 | fnbrfvb 5512 | . . . . . . . 8 | |
13 | 11, 12 | syl5bb 191 | . . . . . . 7 |
14 | 13 | adantll 468 | . . . . . 6 |
15 | 14 | necon3abid 2366 | . . . . 5 |
16 | funfvex 5488 | . . . . . . . 8 | |
17 | 16 | funfni 5273 | . . . . . . 7 |
18 | 17 | adantlr 469 | . . . . . 6 |
19 | breq2 3971 | . . . . . . . 8 | |
20 | 19 | notbid 657 | . . . . . . 7 |
21 | 20 | ceqsexgv 2841 | . . . . . 6 |
22 | 18, 21 | syl 14 | . . . . 5 |
23 | eqcom 2159 | . . . . . . . . . 10 | |
24 | fnbrfvb 5512 | . . . . . . . . . 10 | |
25 | 23, 24 | syl5bb 191 | . . . . . . . . 9 |
26 | 25 | adantlr 469 | . . . . . . . 8 |
27 | 26 | anbi1d 461 | . . . . . . 7 |
28 | brdif 4020 | . . . . . . 7 | |
29 | 27, 28 | bitr4di 197 | . . . . . 6 |
30 | 29 | exbidv 1805 | . . . . 5 |
31 | 15, 22, 30 | 3bitr2rd 216 | . . . 4 |
32 | 10, 31 | syl5bb 191 | . . 3 |
33 | 32 | rabbi2dva 3316 | . 2 |
34 | 8, 33 | eqtr3d 2192 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 wne 2327 crab 2439 cvv 2712 cdif 3099 cin 3101 wss 3102 class class class wbr 3967 cdm 4589 wfn 5168 cfv 5173 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fn 5176 df-fv 5181 |
This theorem is referenced by: fndmdifcom 5576 |
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