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Theorem homfeqd 13614
Description: If two structures have the same  Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homfeqd.1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
homfeqd.2  |-  ( ph  ->  (  Hom  `  C
)  =  (  Hom  `  D ) )
Assertion
Ref Expression
homfeqd  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )

Proof of Theorem homfeqd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homfeqd.2 . . . . 5  |-  ( ph  ->  (  Hom  `  C
)  =  (  Hom  `  D ) )
21oveqd 5891 . . . 4  |-  ( ph  ->  ( x (  Hom  `  C ) y )  =  ( x (  Hom  `  D )
y ) )
32ralrimivw 2640 . . 3  |-  ( ph  ->  A. y  e.  (
Base `  C )
( x (  Hom  `  C ) y )  =  ( x (  Hom  `  D )
y ) )
43ralrimivw 2640 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ( x (  Hom  `  C
) y )  =  ( x (  Hom  `  D ) y ) )
5 eqid 2296 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2296 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
7 eqidd 2297 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  C ) )
8 homfeqd.1 . . 3  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
95, 6, 7, 8homfeq 13613 . 2  |-  ( ph  ->  ( (  Homf  `  C )  =  (  Homf  `  D )  <->  A. x  e.  ( Base `  C ) A. y  e.  ( Base `  C ) ( x (  Hom  `  C
) y )  =  ( x (  Hom  `  D ) y ) ) )
104, 9mpbird 223 1  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235    Homf chomf 13584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-homf 13588
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