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Theorem fcomptf 26124
Description: Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 5981. (Contributed by Thierry Arnoux, 30-Jun-2017.)
Hypothesis
Ref Expression
fcomptf.1  |-  F/_ x B
Assertion
Ref Expression
fcomptf  |-  ( ( A : D --> E  /\  B : C --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
 x ) ) ) )
Distinct variable groups:    x, A    x, C    x, D    x, E
Allowed substitution hint:    B( x)

Proof of Theorem fcomptf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfcv 2613 . . . . 5  |-  F/_ x A
2 nfcv 2613 . . . . 5  |-  F/_ x D
3 nfcv 2613 . . . . 5  |-  F/_ x E
41, 2, 3nff 5656 . . . 4  |-  F/ x  A : D --> E
5 fcomptf.1 . . . . 5  |-  F/_ x B
6 nfcv 2613 . . . . 5  |-  F/_ x C
75, 6, 2nff 5656 . . . 4  |-  F/ x  B : C --> D
84, 7nfan 1863 . . 3  |-  F/ x
( A : D --> E  /\  B : C --> D )
9 ffvelrn 5943 . . . . 5  |-  ( ( B : C --> D  /\  x  e.  C )  ->  ( B `  x
)  e.  D )
109adantll 713 . . . 4  |-  ( ( ( A : D --> E  /\  B : C --> D )  /\  x  e.  C )  ->  ( B `  x )  e.  D )
1110ex 434 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  ( x  e.  C  ->  ( B `  x )  e.  D
) )
128, 11ralrimi 2818 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A. x  e.  C  ( B `  x )  e.  D )
13 ffn 5660 . . . 4  |-  ( B : C --> D  ->  B  Fn  C )
1413adantl 466 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  Fn  C
)
155dffn5f 5848 . . 3  |-  ( B  Fn  C  <->  B  =  ( x  e.  C  |->  ( B `  x
) ) )
1614, 15sylib 196 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  =  ( x  e.  C  |->  ( B `  x ) ) )
17 ffn 5660 . . . 4  |-  ( A : D --> E  ->  A  Fn  D )
1817adantr 465 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  Fn  D
)
19 dffn5 5839 . . 3  |-  ( A  Fn  D  <->  A  =  ( y  e.  D  |->  ( A `  y
) ) )
2018, 19sylib 196 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  =  ( y  e.  D  |->  ( A `  y ) ) )
21 fveq2 5792 . 2  |-  ( y  =  ( B `  x )  ->  ( A `  y )  =  ( A `  ( B `  x ) ) )
2212, 16, 20, 21fmptcof 5979 1  |-  ( ( A : D --> E  /\  B : C --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
 x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   F/_wnfc 2599    |-> cmpt 4451    o. ccom 4945    Fn wfn 5514   -->wf 5515   ` cfv 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527
This theorem is referenced by:  ofoprabco  26126
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