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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 3197 | . . . . 5 | |
2 | dmss 4733 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | fndm 5217 | . . . . 5 | |
5 | 4 | adantr 274 | . . . 4 |
6 | 3, 5 | sseqtrid 3142 | . . 3 |
7 | dfss1 3275 | . . 3 | |
8 | 6, 7 | sylib 121 | . 2 |
9 | vex 2684 | . . . . 5 | |
10 | 9 | eldm 4731 | . . . 4 |
11 | eqcom 2139 | . . . . . . . 8 | |
12 | fnbrfvb 5455 | . . . . . . . 8 | |
13 | 11, 12 | syl5bb 191 | . . . . . . 7 |
14 | 13 | adantll 467 | . . . . . 6 |
15 | 14 | necon3abid 2345 | . . . . 5 |
16 | funfvex 5431 | . . . . . . . 8 | |
17 | 16 | funfni 5218 | . . . . . . 7 |
18 | 17 | adantlr 468 | . . . . . 6 |
19 | breq2 3928 | . . . . . . . 8 | |
20 | 19 | notbid 656 | . . . . . . 7 |
21 | 20 | ceqsexgv 2809 | . . . . . 6 |
22 | 18, 21 | syl 14 | . . . . 5 |
23 | eqcom 2139 | . . . . . . . . . 10 | |
24 | fnbrfvb 5455 | . . . . . . . . . 10 | |
25 | 23, 24 | syl5bb 191 | . . . . . . . . 9 |
26 | 25 | adantlr 468 | . . . . . . . 8 |
27 | 26 | anbi1d 460 | . . . . . . 7 |
28 | brdif 3976 | . . . . . . 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 3279 | . 2 |
34 | 8, 33 | eqtr3d 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wne 2306 crab 2418 cvv 2681 cdif 3063 cin 3065 wss 3066 class class class wbr 3924 cdm 4534 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-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 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-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 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-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: fndmdifcom 5519 |
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