Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 5517 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | offveq.2 | . . . 4 | |
5 | offveq.3 | . . . 4 | |
6 | offveq.1 | . . . 4 | |
7 | inidm 3317 | . . . 4 | |
8 | offveq.5 | . . . 4 | |
9 | offveq.6 | . . . 4 | |
10 | 4, 5, 6, 6, 7, 8, 9 | offval 6042 | . . 3 |
11 | 3, 10 | eqeq12d 2172 | . 2 |
12 | funfvex 5488 | . . . . . 6 | |
13 | 12 | funfni 5273 | . . . . 5 |
14 | 1, 13 | sylan 281 | . . . 4 |
15 | 14 | ralrimiva 2530 | . . 3 |
16 | mpteqb 5561 | . . 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 1335 wcel 2128 wral 2435 cvv 2712 cmpt 4028 wfn 5168 cfv 5173 (class class class)co 5827 cof 6033 |
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 604 ax-in2 605 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-coll 4082 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-setind 4499 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 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-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 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-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-ov 5830 df-oprab 5831 df-mpo 5832 df-of 6035 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |