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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 5219 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | offveq.2 | . . . 4 | |
5 | offveq.3 | . . . 4 | |
6 | offveq.1 | . . . 4 | |
7 | inidm 3146 | . . . 4 | |
8 | offveq.5 | . . . 4 | |
9 | offveq.6 | . . . 4 | |
10 | 4, 5, 6, 6, 7, 8, 9 | offval 5719 | . . 3 |
11 | 3, 10 | eqeq12d 2054 | . 2 |
12 | funfvex 5192 | . . . . . 6 | |
13 | 12 | funfni 4999 | . . . . 5 |
14 | 1, 13 | sylan 267 | . . . 4 |
15 | 14 | ralrimiva 2392 | . . 3 |
16 | mpteqb 5261 | . . 3 | |
17 | 15, 16 | syl 14 | . 2 |
18 | 11, 17 | bitrd 177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wral 2306 cvv 2557 cmpt 3818 wfn 4897 cfv 4902 (class class class)co 5512 cof 5710 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-of 5712 |
This theorem is referenced by: (None) |
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