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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 3253 | . . . . 5 | |
2 | dmss 4810 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | fndm 5297 | . . . . 5 | |
5 | 4 | adantr 274 | . . . 4 |
6 | 3, 5 | sseqtrid 3197 | . . 3 |
7 | dfss1 3331 | . . 3 | |
8 | 6, 7 | sylib 121 | . 2 |
9 | vex 2733 | . . . . 5 | |
10 | 9 | eldm 4808 | . . . 4 |
11 | eqcom 2172 | . . . . . . . 8 | |
12 | fnbrfvb 5537 | . . . . . . . 8 | |
13 | 11, 12 | syl5bb 191 | . . . . . . 7 |
14 | 13 | adantll 473 | . . . . . 6 |
15 | 14 | necon3abid 2379 | . . . . 5 |
16 | funfvex 5513 | . . . . . . . 8 | |
17 | 16 | funfni 5298 | . . . . . . 7 |
18 | 17 | adantlr 474 | . . . . . 6 |
19 | breq2 3993 | . . . . . . . 8 | |
20 | 19 | notbid 662 | . . . . . . 7 |
21 | 20 | ceqsexgv 2859 | . . . . . 6 |
22 | 18, 21 | syl 14 | . . . . 5 |
23 | eqcom 2172 | . . . . . . . . . 10 | |
24 | fnbrfvb 5537 | . . . . . . . . . 10 | |
25 | 23, 24 | syl5bb 191 | . . . . . . . . 9 |
26 | 25 | adantlr 474 | . . . . . . . 8 |
27 | 26 | anbi1d 462 | . . . . . . 7 |
28 | brdif 4042 | . . . . . . 7 | |
29 | 27, 28 | bitr4di 197 | . . . . . 6 |
30 | 29 | exbidv 1818 | . . . . 5 |
31 | 15, 22, 30 | 3bitr2rd 216 | . . . 4 |
32 | 10, 31 | syl5bb 191 | . . 3 |
33 | 32 | rabbi2dva 3335 | . 2 |
34 | 8, 33 | eqtr3d 2205 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 wne 2340 crab 2452 cvv 2730 cdif 3118 cin 3120 wss 3121 class class class wbr 3989 cdm 4611 wfn 5193 cfv 5198 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 |
This theorem is referenced by: fndmdifcom 5602 |
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