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Mirrors > Home > ILE Home > Th. List > mpt2xopovel | 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 |
---|---|
mpt2xopoveq.f |
Ref | Expression |
---|---|
mpt2xopovel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2xopoveq.f | . . . 4 | |
2 | 1 | mpt2xopn0yelv 5854 | . . 3 |
3 | 2 | pm4.71rd 374 | . 2 |
4 | 1 | mpt2xopoveq 5855 | . . . . . 6 |
5 | 4 | eleq2d 2107 | . . . . 5 |
6 | nfcv 2178 | . . . . . . 7 | |
7 | 6 | elrabsf 2801 | . . . . . 6 |
8 | sbccom 2833 | . . . . . . . 8 | |
9 | sbccom 2833 | . . . . . . . . 9 | |
10 | 9 | sbcbii 2818 | . . . . . . . 8 |
11 | 8, 10 | bitri 173 | . . . . . . 7 |
12 | 11 | anbi2i 430 | . . . . . 6 |
13 | 7, 12 | bitri 173 | . . . . 5 |
14 | 5, 13 | syl6bb 185 | . . . 4 |
15 | 14 | pm5.32da 425 | . . 3 |
16 | 3anass 889 | . . 3 | |
17 | 15, 16 | syl6bbr 187 | . 2 |
18 | 3, 17 | bitrd 177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 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-csb 2853 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-iun 3659 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-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: (None) |
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