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Theorem elmapresaunres2 26522
Description: fresaunres2 5556 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)
Assertion
Ref Expression
elmapresaunres2  |-  ( ( F  e.  ( C  ^m  A )  /\  G  e.  ( C  ^m  B )  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  ( ( F  u.  G )  |`  B )  =  G )

Proof of Theorem elmapresaunres2
StepHypRef Expression
1 elmapi 6975 . 2  |-  ( F  e.  ( C  ^m  A )  ->  F : A --> C )
2 elmapi 6975 . 2  |-  ( G  e.  ( C  ^m  B )  ->  G : B --> C )
3 id 20 . 2  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) )  ->  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B
) ) )
4 fresaunres2 5556 . 2  |-  ( ( F : A --> C  /\  G : B --> C  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  (
( F  u.  G
)  |`  B )  =  G )
51, 2, 3, 4syl3an 1226 1  |-  ( ( F  e.  ( C  ^m  A )  /\  G  e.  ( C  ^m  B )  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  ( ( F  u.  G )  |`  B )  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    u. cun 3262    i^i cin 3263    |` cres 4821   -->wf 5391  (class class class)co 6021    ^m cmap 6955
This theorem is referenced by:  diophin  26523  eldioph4b  26564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-map 6957
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