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Theorem fcomptf 27161
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 6048. (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 2622 . . . . 5  |-  F/_ x A
2 nfcv 2622 . . . . 5  |-  F/_ x D
3 nfcv 2622 . . . . 5  |-  F/_ x E
41, 2, 3nff 5718 . . . 4  |-  F/ x  A : D --> E
5 fcomptf.1 . . . . 5  |-  F/_ x B
6 nfcv 2622 . . . . 5  |-  F/_ x C
75, 6, 2nff 5718 . . . 4  |-  F/ x  B : C --> D
84, 7nfan 1870 . . 3  |-  F/ x
( A : D --> E  /\  B : C --> D )
9 ffvelrn 6010 . . . . 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 2857 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A. x  e.  C  ( B `  x )  e.  D )
13 ffn 5722 . . . 4  |-  ( B : C --> D  ->  B  Fn  C )
1413adantl 466 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  Fn  C
)
155dffn5f 5913 . . 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 5722 . . . 4  |-  ( A : D --> E  ->  A  Fn  D )
1817adantr 465 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  Fn  D
)
19 dffn5 5904 . . 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 5857 . 2  |-  ( y  =  ( B `  x )  ->  ( A `  y )  =  ( A `  ( B `  x ) ) )
2212, 16, 20, 21fmptcof 6046 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 1374    e. wcel 1762   F/_wnfc 2608    |-> cmpt 4498    o. ccom 4996    Fn wfn 5574   -->wf 5575   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587
This theorem is referenced by:  ofoprabco  27163
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