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Mirrors > Home > ILE Home > Th. List > fneqeql2 | Unicode version |
Description: Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
fneqeql2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneqeql 5521 | . 2 | |
2 | inss1 3291 | . . . . . 6 | |
3 | dmss 4733 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | fndm 5217 | . . . . . 6 | |
6 | 5 | adantr 274 | . . . . 5 |
7 | 4, 6 | sseqtrid 3142 | . . . 4 |
8 | 7 | biantrurd 303 | . . 3 |
9 | eqss 3107 | . . 3 | |
10 | 8, 9 | syl6rbbr 198 | . 2 |
11 | 1, 10 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 cin 3065 wss 3066 cdm 4534 wfn 5113 |
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 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-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 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-mpt 3986 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: (None) |
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