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Mirrors > Home > ILE Home > Th. List > isocnv2 | Unicode version |
Description: Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
Ref | Expression |
---|---|
isocnv2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isof1o 5757 | . . 3 | |
2 | f1ofn 5415 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | isof1o 5757 | . . 3 | |
5 | 4, 2 | syl 14 | . 2 |
6 | ralcom 2620 | . . . . 5 | |
7 | vex 2715 | . . . . . . . . . 10 | |
8 | vex 2715 | . . . . . . . . . 10 | |
9 | 7, 8 | brcnv 4769 | . . . . . . . . 9 |
10 | 9 | a1i 9 | . . . . . . . 8 |
11 | funfvex 5485 | . . . . . . . . . . 11 | |
12 | 11 | funfni 5270 | . . . . . . . . . 10 |
13 | 12 | adantr 274 | . . . . . . . . 9 |
14 | funfvex 5485 | . . . . . . . . . . 11 | |
15 | 14 | funfni 5270 | . . . . . . . . . 10 |
16 | 15 | adantlr 469 | . . . . . . . . 9 |
17 | brcnvg 4767 | . . . . . . . . 9 | |
18 | 13, 16, 17 | syl2anc 409 | . . . . . . . 8 |
19 | 10, 18 | bibi12d 234 | . . . . . . 7 |
20 | 19 | ralbidva 2453 | . . . . . 6 |
21 | 20 | ralbidva 2453 | . . . . 5 |
22 | 6, 21 | bitr4id 198 | . . . 4 |
23 | 22 | anbi2d 460 | . . 3 |
24 | df-isom 5179 | . . 3 | |
25 | df-isom 5179 | . . 3 | |
26 | 23, 24, 25 | 3bitr4g 222 | . 2 |
27 | 3, 5, 26 | pm5.21nii 694 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wcel 2128 wral 2435 cvv 2712 class class class wbr 3965 ccnv 4585 wfn 5165 wf1o 5169 cfv 5170 wiso 5171 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-f1o 5177 df-fv 5178 df-isom 5179 |
This theorem is referenced by: infisoti 6976 |
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