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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 5542 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | offveq.2 | . . . 4 | |
5 | offveq.3 | . . . 4 | |
6 | offveq.1 | . . . 4 | |
7 | inidm 3336 | . . . 4 | |
8 | offveq.5 | . . . 4 | |
9 | offveq.6 | . . . 4 | |
10 | 4, 5, 6, 6, 7, 8, 9 | offval 6068 | . . 3 |
11 | 3, 10 | eqeq12d 2185 | . 2 |
12 | funfvex 5513 | . . . . . 6 | |
13 | 12 | funfni 5298 | . . . . 5 |
14 | 1, 13 | sylan 281 | . . . 4 |
15 | 14 | ralrimiva 2543 | . . 3 |
16 | mpteqb 5586 | . . 3 | |
17 | 15, 16 | syl 14 | . 2 |
18 | 11, 17 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 cvv 2730 cmpt 4050 wfn 5193 cfv 5198 (class class class)co 5853 cof 6059 |
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-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 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-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-of 6061 |
This theorem is referenced by: (None) |
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