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Mirrors > Home > ILE Home > Th. List > cnvf1olem | Unicode version |
Description: Lemma for cnvf1o 5846. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
cnvf1olem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 484 | . . . 4 | |
2 | 1st2nd 5807 | . . . . . . . 8 | |
3 | 2 | adantrr 448 | . . . . . . 7 |
4 | 3 | sneqd 3388 | . . . . . 6 |
5 | 4 | cnveqd 4511 | . . . . 5 |
6 | 5 | unieqd 3591 | . . . 4 |
7 | 1stexg 5794 | . . . . . 6 | |
8 | 2ndexg 5795 | . . . . . 6 | |
9 | opswapg 4807 | . . . . . 6 | |
10 | 7, 8, 9 | syl2anc 391 | . . . . 5 |
11 | 10 | ad2antrl 459 | . . . 4 |
12 | 1, 6, 11 | 3eqtrd 2076 | . . 3 |
13 | simprl 483 | . . . . 5 | |
14 | 3, 13 | eqeltrrd 2115 | . . . 4 |
15 | opelcnvg 4515 | . . . . . 6 | |
16 | 8, 7, 15 | syl2anc 391 | . . . . 5 |
17 | 16 | ad2antrl 459 | . . . 4 |
18 | 14, 17 | mpbird 156 | . . 3 |
19 | 12, 18 | eqeltrd 2114 | . 2 |
20 | opswapg 4807 | . . . . . 6 | |
21 | 8, 7, 20 | syl2anc 391 | . . . . 5 |
22 | 21 | eqcomd 2045 | . . . 4 |
23 | 22 | ad2antrl 459 | . . 3 |
24 | 12 | sneqd 3388 | . . . . 5 |
25 | 24 | cnveqd 4511 | . . . 4 |
26 | 25 | unieqd 3591 | . . 3 |
27 | 23, 3, 26 | 3eqtr4d 2082 | . 2 |
28 | 19, 27 | jca 290 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 cvv 2557 csn 3375 cop 3378 cuni 3580 ccnv 4344 wrel 4350 cfv 4902 c1st 5765 c2nd 5766 |
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-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 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 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-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 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-fn 4905 df-f 4906 df-fo 4908 df-fv 4910 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: cnvf1o 5846 |
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