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Mirrors > Home > ILE Home > Th. List > fnreseql | Unicode version |
Description: Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
fnreseql |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnssres 5285 | . . . 4 | |
2 | 1 | 3adant2 1001 | . . 3 |
3 | fnssres 5285 | . . . 4 | |
4 | 3 | 3adant1 1000 | . . 3 |
5 | fneqeql 5577 | . . 3 | |
6 | 2, 4, 5 | syl2anc 409 | . 2 |
7 | resindir 4884 | . . . . . 6 | |
8 | 7 | dmeqi 4789 | . . . . 5 |
9 | dmres 4889 | . . . . 5 | |
10 | 8, 9 | eqtr3i 2180 | . . . 4 |
11 | 10 | eqeq1i 2165 | . . 3 |
12 | df-ss 3115 | . . 3 | |
13 | 11, 12 | bitr4i 186 | . 2 |
14 | 6, 13 | bitrdi 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 963 wceq 1335 cin 3101 wss 3102 cdm 4588 cres 4590 wfn 5167 |
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-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 4084 ax-pow 4137 ax-pr 4171 |
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-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 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 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-res 4600 df-iota 5137 df-fun 5174 df-fn 5175 df-fv 5180 |
This theorem is referenced by: (None) |
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