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Mirrors > Home > MPE Home > Th. List > 2fvcoidd | Structured version Visualization version Unicode version |
Description: Show that the composition of two functions is the identity function by applying both functions to each value of the domain of the first function. (Contributed by AV, 15-Dec-2019.) |
Ref | Expression |
---|---|
2fvcoidd.f | |
2fvcoidd.g | |
2fvcoidd.i |
Ref | Expression |
---|---|
2fvcoidd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fvcoidd.g | . . 3 | |
2 | 2fvcoidd.f | . . 3 | |
3 | fcompt 6400 | . . 3 | |
4 | 1, 2, 3 | syl2anc 693 | . 2 |
5 | 2fvcoidd.i | . . . . . 6 | |
6 | fveq2 6191 | . . . . . . . . 9 | |
7 | 6 | fveq2d 6195 | . . . . . . . 8 |
8 | id 22 | . . . . . . . 8 | |
9 | 7, 8 | eqeq12d 2637 | . . . . . . 7 |
10 | 9 | rspccv 3306 | . . . . . 6 |
11 | 5, 10 | syl 17 | . . . . 5 |
12 | 11 | imp 445 | . . . 4 |
13 | 12 | mpteq2dva 4744 | . . 3 |
14 | mptresid 5456 | . . 3 | |
15 | 13, 14 | syl6eq 2672 | . 2 |
16 | 4, 15 | eqtrd 2656 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 wral 2912 cmpt 4729 cid 5023 cres 5116 ccom 5118 wf 5884 cfv 5888 |
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-8 1992 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-pow 4843 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-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 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-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 |
This theorem is referenced by: 2fvidf1od 6553 2fvidinvd 6554 |
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