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Mirrors > Home > ILE Home > Th. List > offveqb | Unicode version |
Description: Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
Ref | Expression |
---|---|
offveq.1 | |
offveq.2 | |
offveq.3 | |
offveq.4 | |
offveq.5 | |
offveq.6 |
Ref | Expression |
---|---|
offveqb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offveq.4 | . . . 4 | |
2 | dffn5im 5240 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | offveq.2 | . . . 4 | |
5 | offveq.3 | . . . 4 | |
6 | offveq.1 | . . . 4 | |
7 | inidm 3175 | . . . 4 | |
8 | offveq.5 | . . . 4 | |
9 | offveq.6 | . . . 4 | |
10 | 4, 5, 6, 6, 7, 8, 9 | offval 5739 | . . 3 |
11 | 3, 10 | eqeq12d 2095 | . 2 |
12 | funfvex 5212 | . . . . . 6 | |
13 | 12 | funfni 5019 | . . . . 5 |
14 | 1, 13 | sylan 277 | . . . 4 |
15 | 14 | ralrimiva 2434 | . . 3 |
16 | mpteqb 5282 | . . 3 | |
17 | 15, 16 | syl 14 | . 2 |
18 | 11, 17 | bitrd 186 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 cvv 2601 cmpt 3839 wfn 4917 cfv 4922 (class class class)co 5532 cof 5730 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-of 5732 |
This theorem is referenced by: (None) |
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