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Theorem brcoffn 38328
Description: Conditions allowing the decomposition of a binary relation. (Contributed by RP, 7-Jun-2021.)
Hypotheses
Ref Expression
brcoffn.c  |-  ( ph  ->  C  Fn  Y )
brcoffn.d  |-  ( ph  ->  D : X --> Y )
brcoffn.r  |-  ( ph  ->  A ( C  o.  D ) B )
Assertion
Ref Expression
brcoffn  |-  ( ph  ->  ( A D ( D `  A )  /\  ( D `  A ) C B ) )

Proof of Theorem brcoffn
StepHypRef Expression
1 brcoffn.c . . . 4  |-  ( ph  ->  C  Fn  Y )
2 brcoffn.d . . . 4  |-  ( ph  ->  D : X --> Y )
3 fnfco 6069 . . . 4  |-  ( ( C  Fn  Y  /\  D : X --> Y )  ->  ( C  o.  D )  Fn  X
)
41, 2, 3syl2anc 693 . . 3  |-  ( ph  ->  ( C  o.  D
)  Fn  X )
5 simpl 473 . . . 4  |-  ( (
ph  /\  ( C  o.  D )  Fn  X
)  ->  ph )
6 simpr 477 . . . 4  |-  ( (
ph  /\  ( C  o.  D )  Fn  X
)  ->  ( C  o.  D )  Fn  X
)
7 brcoffn.r . . . . . 6  |-  ( ph  ->  A ( C  o.  D ) B )
85, 7syl 17 . . . . 5  |-  ( (
ph  /\  ( C  o.  D )  Fn  X
)  ->  A ( C  o.  D ) B )
9 fnbr 5993 . . . . 5  |-  ( ( ( C  o.  D
)  Fn  X  /\  A ( C  o.  D ) B )  ->  A  e.  X
)
106, 8, 9syl2anc 693 . . . 4  |-  ( (
ph  /\  ( C  o.  D )  Fn  X
)  ->  A  e.  X )
115, 6, 103jca 1242 . . 3  |-  ( (
ph  /\  ( C  o.  D )  Fn  X
)  ->  ( ph  /\  ( C  o.  D
)  Fn  X  /\  A  e.  X )
)
124, 11mpdan 702 . 2  |-  ( ph  ->  ( ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X ) )
1323ad2ant1 1082 . . . . . 6  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  D : X
--> Y )
14 simp3 1063 . . . . . 6  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  A  e.  X )
15 fvco3 6275 . . . . . 6  |-  ( ( D : X --> Y  /\  A  e.  X )  ->  ( ( C  o.  D ) `  A
)  =  ( C `
 ( D `  A ) ) )
1613, 14, 15syl2anc 693 . . . . 5  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( ( C  o.  D ) `  A )  =  ( C `  ( D `
 A ) ) )
1773ad2ant1 1082 . . . . . 6  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  A ( C  o.  D ) B )
18 fnbrfvb 6236 . . . . . . 7  |-  ( ( ( C  o.  D
)  Fn  X  /\  A  e.  X )  ->  ( ( ( C  o.  D ) `  A )  =  B  <-> 
A ( C  o.  D ) B ) )
19183adant1 1079 . . . . . 6  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( (
( C  o.  D
) `  A )  =  B  <->  A ( C  o.  D ) B ) )
2017, 19mpbird 247 . . . . 5  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( ( C  o.  D ) `  A )  =  B )
2116, 20eqtr3d 2658 . . . 4  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( C `  ( D `  A
) )  =  B )
22 eqid 2622 . . . 4  |-  ( D `
 A )  =  ( D `  A
)
2321, 22jctil 560 . . 3  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( ( D `  A )  =  ( D `  A )  /\  ( C `  ( D `  A ) )  =  B ) )
2413ffnd 6046 . . . . 5  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  D  Fn  X )
25 fnbrfvb 6236 . . . . 5  |-  ( ( D  Fn  X  /\  A  e.  X )  ->  ( ( D `  A )  =  ( D `  A )  <-> 
A D ( D `
 A ) ) )
2624, 14, 25syl2anc 693 . . . 4  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( ( D `  A )  =  ( D `  A )  <->  A D
( D `  A
) ) )
2713ad2ant1 1082 . . . . 5  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  C  Fn  Y )
2813, 14ffvelrnd 6360 . . . . 5  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( D `  A )  e.  Y
)
29 fnbrfvb 6236 . . . . 5  |-  ( ( C  Fn  Y  /\  ( D `  A )  e.  Y )  -> 
( ( C `  ( D `  A ) )  =  B  <->  ( D `  A ) C B ) )
3027, 28, 29syl2anc 693 . . . 4  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( ( C `  ( D `  A ) )  =  B  <->  ( D `  A ) C B ) )
3126, 30anbi12d 747 . . 3  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( (
( D `  A
)  =  ( D `
 A )  /\  ( C `  ( D `
 A ) )  =  B )  <->  ( A D ( D `  A )  /\  ( D `  A ) C B ) ) )
3223, 31mpbid 222 . 2  |-  ( (
ph  /\  ( C  o.  D )  Fn  X  /\  A  e.  X
)  ->  ( A D ( D `  A )  /\  ( D `  A ) C B ) )
3312, 32syl 17 1  |-  ( ph  ->  ( A D ( D `  A )  /\  ( D `  A ) C B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 196    /\ wa 384    /\ w3a 1037    = wceq 1483    e. wcel 1990   class class class wbr 4653    o. ccom 5118    Fn wfn 5883   -->wf 5884   ` cfv 5888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896
This theorem is referenced by:  brcofffn  38329  brco2f1o  38330  clsneikex  38404  clsneinex  38405  clsneiel1  38406
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