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Theorem cofunex2g 6739
Description: Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cofunex2g  |-  ( ( A  e.  V  /\  Fun  `' B )  ->  ( A  o.  B )  e.  _V )

Proof of Theorem cofunex2g
StepHypRef Expression
1 cnvexg 6720 . . . 4  |-  ( A  e.  V  ->  `' A  e.  _V )
2 cofunexg 6738 . . . 4  |-  ( ( Fun  `' B  /\  `' A  e.  _V )  ->  ( `' B  o.  `' A )  e.  _V )
31, 2sylan2 474 . . 3  |-  ( ( Fun  `' B  /\  A  e.  V )  ->  ( `' B  o.  `' A )  e.  _V )
4 cnvco 5179 . . . . 5  |-  `' ( `' B  o.  `' A )  =  ( `' `' A  o.  `' `' B )
5 cocnvcnv2 5510 . . . . 5  |-  ( `' `' A  o.  `' `' B )  =  ( `' `' A  o.  B
)
6 cocnvcnv1 5509 . . . . 5  |-  ( `' `' A  o.  B
)  =  ( A  o.  B )
74, 5, 63eqtrri 2494 . . . 4  |-  ( A  o.  B )  =  `' ( `' B  o.  `' A )
8 cnvexg 6720 . . . 4  |-  ( ( `' B  o.  `' A )  e.  _V  ->  `' ( `' B  o.  `' A )  e.  _V )
97, 8syl5eqel 2552 . . 3  |-  ( ( `' B  o.  `' A )  e.  _V  ->  ( A  o.  B
)  e.  _V )
103, 9syl 16 . 2  |-  ( ( Fun  `' B  /\  A  e.  V )  ->  ( A  o.  B
)  e.  _V )
1110ancoms 453 1  |-  ( ( A  e.  V  /\  Fun  `' B )  ->  ( A  o.  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   _Vcvv 3106   `'ccnv 4991    o. ccom 4996   Fun wfun 5573
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-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
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-reu 2814  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-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  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-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587
This theorem is referenced by:  fsuppco  7850
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