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Mirrors > Home > ILE Home > Th. List > cnvf1olem | Unicode version |
Description: Lemma for cnvf1o 6184. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
cnvf1olem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 522 | . . . 4 | |
2 | 1st2nd 6141 | . . . . . . . 8 | |
3 | 2 | adantrr 471 | . . . . . . 7 |
4 | 3 | sneqd 3583 | . . . . . 6 |
5 | 4 | cnveqd 4774 | . . . . 5 |
6 | 5 | unieqd 3794 | . . . 4 |
7 | 1stexg 6127 | . . . . . 6 | |
8 | 2ndexg 6128 | . . . . . 6 | |
9 | opswapg 5084 | . . . . . 6 | |
10 | 7, 8, 9 | syl2anc 409 | . . . . 5 |
11 | 10 | ad2antrl 482 | . . . 4 |
12 | 1, 6, 11 | 3eqtrd 2201 | . . 3 |
13 | simprl 521 | . . . . 5 | |
14 | 3, 13 | eqeltrrd 2242 | . . . 4 |
15 | opelcnvg 4778 | . . . . . 6 | |
16 | 8, 7, 15 | syl2anc 409 | . . . . 5 |
17 | 16 | ad2antrl 482 | . . . 4 |
18 | 14, 17 | mpbird 166 | . . 3 |
19 | 12, 18 | eqeltrd 2241 | . 2 |
20 | opswapg 5084 | . . . . . 6 | |
21 | 8, 7, 20 | syl2anc 409 | . . . . 5 |
22 | 21 | eqcomd 2170 | . . . 4 |
23 | 22 | ad2antrl 482 | . . 3 |
24 | 12 | sneqd 3583 | . . . . 5 |
25 | 24 | cnveqd 4774 | . . . 4 |
26 | 25 | unieqd 3794 | . . 3 |
27 | 23, 3, 26 | 3eqtr4d 2207 | . 2 |
28 | 19, 27 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cvv 2721 csn 3570 cop 3573 cuni 3783 ccnv 4597 wrel 4603 cfv 5182 c1st 6098 c2nd 6099 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 df-1st 6100 df-2nd 6101 |
This theorem is referenced by: cnvf1o 6184 |
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