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Mirrors > Home > ILE Home > Th. List > mpoxopovel | Unicode version |
Description: Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopoveq.f |
Ref | Expression |
---|---|
mpoxopovel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxopoveq.f | . . . 4 | |
2 | 1 | mpoxopn0yelv 6230 | . . 3 |
3 | 2 | pm4.71rd 394 | . 2 |
4 | 1 | mpoxopoveq 6231 | . . . . . 6 |
5 | 4 | eleq2d 2245 | . . . . 5 |
6 | nfcv 2317 | . . . . . . 7 | |
7 | 6 | elrabsf 2999 | . . . . . 6 |
8 | sbccom 3036 | . . . . . . . 8 | |
9 | sbccom 3036 | . . . . . . . . 9 | |
10 | 9 | sbcbii 3020 | . . . . . . . 8 |
11 | 8, 10 | bitri 184 | . . . . . . 7 |
12 | 11 | anbi2i 457 | . . . . . 6 |
13 | 7, 12 | bitri 184 | . . . . 5 |
14 | 5, 13 | bitrdi 196 | . . . 4 |
15 | 14 | pm5.32da 452 | . . 3 |
16 | 3anass 982 | . . 3 | |
17 | 15, 16 | bitr4di 198 | . 2 |
18 | 3, 17 | bitrd 188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 w3a 978 wceq 1353 wcel 2146 crab 2457 cvv 2735 wsbc 2960 cop 3592 cfv 5208 (class class class)co 5865 cmpo 5867 c1st 6129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 |
This theorem is referenced by: (None) |
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