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Mirrors > Home > ILE Home > Th. List > funcocnv2 | Unicode version |
Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
funcocnv2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fun 5120 | . . 3 | |
2 | 1 | simprbi 273 | . 2 |
3 | iss 4860 | . . 3 | |
4 | dfdm4 4726 | . . . . . . . 8 | |
5 | dmcoeq 4806 | . . . . . . . 8 | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 |
7 | df-rn 4545 | . . . . . . 7 | |
8 | 6, 7 | eqtr4i 2161 | . . . . . 6 |
9 | 8 | a1i 9 | . . . . 5 |
10 | 9 | reseq2d 4814 | . . . 4 |
11 | 10 | eqeq2d 2149 | . . 3 |
12 | 3, 11 | syl5bb 191 | . 2 |
13 | 2, 12 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wss 3066 cid 4205 ccnv 4533 cdm 4534 crn 4535 cres 4536 ccom 4538 wrel 4539 wfun 5112 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-fun 5120 |
This theorem is referenced by: fococnv2 5386 f1cocnv2 5388 funcoeqres 5391 |
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