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Mirrors > Home > MPE Home > Th. List > funcocnv2 | Structured version Visualization version Unicode version |
Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
funcocnv2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fun 5890 | . . 3 | |
2 | 1 | simprbi 480 | . 2 |
3 | iss 5447 | . . 3 | |
4 | dfdm4 5316 | . . . . . . 7 | |
5 | dmcoeq 5388 | . . . . . . 7 | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 |
7 | df-rn 5125 | . . . . . 6 | |
8 | 6, 7 | eqtr4i 2647 | . . . . 5 |
9 | 8 | reseq2i 5393 | . . . 4 |
10 | 9 | eqeq2i 2634 | . . 3 |
11 | 3, 10 | bitri 264 | . 2 |
12 | 2, 11 | sylib 208 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wss 3574 cid 5023 ccnv 5113 cdm 5114 crn 5115 cres 5116 ccom 5118 wrel 5119 wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-fun 5890 |
This theorem is referenced by: fococnv2 6162 f1cocnv2 6164 funcoeqres 6167 fcoinver 29418 cocnv 33520 frege131d 38056 |
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