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Mirrors > Home > ILE Home > Th. List > mpt2xopoveq | 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 |
---|---|
mpt2xopoveq.f |
Ref | Expression |
---|---|
mpt2xopoveq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2xopoveq.f | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | fveq2 5178 | . . . . 5 | |
4 | op1stg 5777 | . . . . . 6 | |
5 | 4 | adantr 261 | . . . . 5 |
6 | 3, 5 | sylan9eqr 2094 | . . . 4 |
7 | 6 | adantrr 448 | . . 3 |
8 | sbceq1a 2773 | . . . . . 6 | |
9 | 8 | adantl 262 | . . . . 5 |
10 | 9 | adantl 262 | . . . 4 |
11 | sbceq1a 2773 | . . . . . 6 | |
12 | 11 | adantr 261 | . . . . 5 |
13 | 12 | adantl 262 | . . . 4 |
14 | 10, 13 | bitrd 177 | . . 3 |
15 | 7, 14 | rabeqbidv 2552 | . 2 |
16 | opexg 3964 | . . 3 | |
17 | 16 | adantr 261 | . 2 |
18 | simpr 103 | . 2 | |
19 | rabexg 3900 | . . 3 | |
20 | 19 | ad2antrr 457 | . 2 |
21 | equid 1589 | . . 3 | |
22 | nfvd 1422 | . . 3 | |
23 | 21, 22 | ax-mp 7 | . 2 |
24 | nfvd 1422 | . . 3 | |
25 | 21, 24 | ax-mp 7 | . 2 |
26 | nfcv 2178 | . 2 | |
27 | nfcv 2178 | . 2 | |
28 | nfsbc1v 2782 | . . 3 | |
29 | nfcv 2178 | . . 3 | |
30 | 28, 29 | nfrabxy 2490 | . 2 |
31 | nfsbc1v 2782 | . . . 4 | |
32 | 26, 31 | nfsbc 2784 | . . 3 |
33 | nfcv 2178 | . . 3 | |
34 | 32, 33 | nfrabxy 2490 | . 2 |
35 | 2, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34 | ovmpt2dxf 5626 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wnf 1349 wcel 1393 crab 2310 cvv 2557 wsbc 2764 cop 3378 cfv 4902 (class class class)co 5512 cmpt2 5514 c1st 5765 |
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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 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-rab 2315 df-v 2559 df-sbc 2765 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-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-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 |
This theorem is referenced by: mpt2xopovel 5856 |
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