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Mirrors > Home > ILE Home > Th. List > mpoxopoveq | Unicode version |
Description: 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, 11-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopoveq.f |
Ref | Expression |
---|---|
mpoxopoveq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxopoveq.f | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | fveq2 5421 | . . . . 5 | |
4 | op1stg 6048 | . . . . . 6 | |
5 | 4 | adantr 274 | . . . . 5 |
6 | 3, 5 | sylan9eqr 2194 | . . . 4 |
7 | 6 | adantrr 470 | . . 3 |
8 | sbceq1a 2918 | . . . . . 6 | |
9 | 8 | adantl 275 | . . . . 5 |
10 | 9 | adantl 275 | . . . 4 |
11 | sbceq1a 2918 | . . . . . 6 | |
12 | 11 | adantr 274 | . . . . 5 |
13 | 12 | adantl 275 | . . . 4 |
14 | 10, 13 | bitrd 187 | . . 3 |
15 | 7, 14 | rabeqbidv 2681 | . 2 |
16 | opexg 4150 | . . 3 | |
17 | 16 | adantr 274 | . 2 |
18 | simpr 109 | . 2 | |
19 | rabexg 4071 | . . 3 | |
20 | 19 | ad2antrr 479 | . 2 |
21 | equid 1677 | . . 3 | |
22 | nfvd 1509 | . . 3 | |
23 | 21, 22 | ax-mp 5 | . 2 |
24 | nfvd 1509 | . . 3 | |
25 | 21, 24 | ax-mp 5 | . 2 |
26 | nfcv 2281 | . 2 | |
27 | nfcv 2281 | . 2 | |
28 | nfsbc1v 2927 | . . 3 | |
29 | nfcv 2281 | . . 3 | |
30 | 28, 29 | nfrabxy 2611 | . 2 |
31 | nfsbc1v 2927 | . . . 4 | |
32 | 26, 31 | nfsbc 2929 | . . 3 |
33 | nfcv 2281 | . . 3 | |
34 | 32, 33 | nfrabxy 2611 | . 2 |
35 | 2, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34 | ovmpodxf 5896 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wnf 1436 wcel 1480 crab 2420 cvv 2686 wsbc 2909 cop 3530 cfv 5123 (class class class)co 5774 cmpo 5776 c1st 6036 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 |
This theorem is referenced by: mpoxopovel 6138 |
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