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Theorem 2fvcoidd 6175
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.)
Hypotheses
Ref Expression
2fvcoidd.f  |-  ( ph  ->  F : A --> B )
2fvcoidd.g  |-  ( ph  ->  G : B --> A )
2fvcoidd.i  |-  ( ph  ->  A. a  e.  A  ( G `  ( F `
 a ) )  =  a )
Assertion
Ref Expression
2fvcoidd  |-  ( ph  ->  ( G  o.  F
)  =  (  _I  |`  A ) )
Distinct variable groups:    A, a    F, a    G, a
Allowed substitution hints:    ph( a)    B( a)

Proof of Theorem 2fvcoidd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 2fvcoidd.g . . 3  |-  ( ph  ->  G : B --> A )
2 2fvcoidd.f . . 3  |-  ( ph  ->  F : A --> B )
3 fcompt 6043 . . 3  |-  ( ( G : B --> A  /\  F : A --> B )  ->  ( G  o.  F )  =  ( x  e.  A  |->  ( G `  ( F `
 x ) ) ) )
41, 2, 3syl2anc 659 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  ( G `
 ( F `  x ) ) ) )
5 2fvcoidd.i . . . . . 6  |-  ( ph  ->  A. a  e.  A  ( G `  ( F `
 a ) )  =  a )
6 fveq2 5848 . . . . . . . . 9  |-  ( a  =  x  ->  ( F `  a )  =  ( F `  x ) )
76fveq2d 5852 . . . . . . . 8  |-  ( a  =  x  ->  ( G `  ( F `  a ) )  =  ( G `  ( F `  x )
) )
8 id 22 . . . . . . . 8  |-  ( a  =  x  ->  a  =  x )
97, 8eqeq12d 2476 . . . . . . 7  |-  ( a  =  x  ->  (
( G `  ( F `  a )
)  =  a  <->  ( G `  ( F `  x
) )  =  x ) )
109rspccv 3204 . . . . . 6  |-  ( A. a  e.  A  ( G `  ( F `  a ) )  =  a  ->  ( x  e.  A  ->  ( G `
 ( F `  x ) )  =  x ) )
115, 10syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  ( G `  ( F `  x )
)  =  x ) )
1211imp 427 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  ( F `  x ) )  =  x )
1312mpteq2dva 4525 . . 3  |-  ( ph  ->  ( x  e.  A  |->  ( G `  ( F `  x )
) )  =  ( x  e.  A  |->  x ) )
14 mptresid 5316 . . 3  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
1513, 14syl6eq 2511 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( G `  ( F `  x )
) )  =  (  _I  |`  A )
)
164, 15eqtrd 2495 1  |-  ( ph  ->  ( G  o.  F
)  =  (  _I  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   A.wral 2804    |-> cmpt 4497    _I cid 4779    |` cres 4990    o. ccom 4992   -->wf 5566   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by:  2fvidf1od  6176  2fvidinvd  6177
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