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Theorem fmptcof2 27174
Description: Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypotheses
Ref Expression
fmptcof2.1  |-  F/_ x A
fmptcof2.2  |-  F/_ x B
fmptcof2.3  |-  F/ x ph
fmptcof2.4  |-  ( ph  ->  A. x  e.  A  R  e.  B )
fmptcof2.5  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
fmptcof2.6  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
fmptcof2.7  |-  ( y  =  R  ->  S  =  T )
Assertion
Ref Expression
fmptcof2  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  T ) )
Distinct variable groups:    x, y    y, B    y, R    x, S    y, T
Allowed substitution hints:    ph( x, y)    A( x, y)    B( x)    R( x)    S( y)    T( x)    F( x, y)    G( x, y)

Proof of Theorem fmptcof2
Dummy variables  v  u  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5503 . 2  |-  Rel  ( G  o.  F )
2 funmpt 5622 . . 3  |-  Fun  (
x  e.  A  |->  T )
3 funrel 5603 . . 3  |-  ( Fun  ( x  e.  A  |->  T )  ->  Rel  ( x  e.  A  |->  T ) )
42, 3ax-mp 5 . 2  |-  Rel  (
x  e.  A  |->  T )
5 fmptcof2.3 . . . . . . . . . . . . 13  |-  F/ x ph
6 fmptcof2.1 . . . . . . . . . . . . 13  |-  F/_ x A
7 fmptcof2.2 . . . . . . . . . . . . 13  |-  F/_ x B
8 fmptcof2.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  A  R  e.  B )
98r19.21bi 2833 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  R  e.  B )
10 eqid 2467 . . . . . . . . . . . . 13  |-  ( x  e.  A  |->  R )  =  ( x  e.  A  |->  R )
115, 6, 7, 9, 10fmptdF 27167 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  A  |->  R ) : A --> B )
12 fmptcof2.5 . . . . . . . . . . . . 13  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
1312feq1d 5715 . . . . . . . . . . . 12  |-  ( ph  ->  ( F : A --> B 
<->  ( x  e.  A  |->  R ) : A --> B ) )
1411, 13mpbird 232 . . . . . . . . . . 11  |-  ( ph  ->  F : A --> B )
15 ffun 5731 . . . . . . . . . . 11  |-  ( F : A --> B  ->  Fun  F )
1614, 15syl 16 . . . . . . . . . 10  |-  ( ph  ->  Fun  F )
17 funbrfv 5904 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( z F u  ->  ( F `
 z )  =  u ) )
1817imp 429 . . . . . . . . . 10  |-  ( ( Fun  F  /\  z F u )  -> 
( F `  z
)  =  u )
1916, 18sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  z F u )  ->  ( F `  z )  =  u )
2019eqcomd 2475 . . . . . . . 8  |-  ( (
ph  /\  z F u )  ->  u  =  ( F `  z ) )
2120a1d 25 . . . . . . 7  |-  ( (
ph  /\  z F u )  ->  (
u G w  ->  u  =  ( F `  z ) ) )
2221expimpd 603 . . . . . 6  |-  ( ph  ->  ( ( z F u  /\  u G w )  ->  u  =  ( F `  z ) ) )
2322pm4.71rd 635 . . . . 5  |-  ( ph  ->  ( ( z F u  /\  u G w )  <->  ( u  =  ( F `  z )  /\  (
z F u  /\  u G w ) ) ) )
2423exbidv 1690 . . . 4  |-  ( ph  ->  ( E. u ( z F u  /\  u G w )  <->  E. u
( u  =  ( F `  z )  /\  ( z F u  /\  u G w ) ) ) )
25 fvex 5874 . . . . . 6  |-  ( F `
 z )  e. 
_V
26 breq2 4451 . . . . . . 7  |-  ( u  =  ( F `  z )  ->  (
z F u  <->  z F
( F `  z
) ) )
27 breq1 4450 . . . . . . 7  |-  ( u  =  ( F `  z )  ->  (
u G w  <->  ( F `  z ) G w ) )
2826, 27anbi12d 710 . . . . . 6  |-  ( u  =  ( F `  z )  ->  (
( z F u  /\  u G w )  <->  ( z F ( F `  z
)  /\  ( F `  z ) G w ) ) )
2925, 28ceqsexv 3150 . . . . 5  |-  ( E. u ( u  =  ( F `  z
)  /\  ( z F u  /\  u G w ) )  <-> 
( z F ( F `  z )  /\  ( F `  z ) G w ) )
30 funfvbrb 5992 . . . . . . . . 9  |-  ( Fun 
F  ->  ( z  e.  dom  F  <->  z F
( F `  z
) ) )
3116, 30syl 16 . . . . . . . 8  |-  ( ph  ->  ( z  e.  dom  F  <-> 
z F ( F `
 z ) ) )
32 fdm 5733 . . . . . . . . . 10  |-  ( F : A --> B  ->  dom  F  =  A )
3314, 32syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  A )
3433eleq2d 2537 . . . . . . . 8  |-  ( ph  ->  ( z  e.  dom  F  <-> 
z  e.  A ) )
3531, 34bitr3d 255 . . . . . . 7  |-  ( ph  ->  ( z F ( F `  z )  <-> 
z  e.  A ) )
3612fveq1d 5866 . . . . . . . 8  |-  ( ph  ->  ( F `  z
)  =  ( ( x  e.  A  |->  R ) `  z ) )
37 fmptcof2.6 . . . . . . . 8  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
38 eqidd 2468 . . . . . . . 8  |-  ( ph  ->  w  =  w )
3936, 37, 38breq123d 4461 . . . . . . 7  |-  ( ph  ->  ( ( F `  z ) G w  <-> 
( ( x  e.  A  |->  R ) `  z ) ( y  e.  B  |->  S ) w ) )
4035, 39anbi12d 710 . . . . . 6  |-  ( ph  ->  ( ( z F ( F `  z
)  /\  ( F `  z ) G w )  <->  ( z  e.  A  /\  ( ( x  e.  A  |->  R ) `  z ) ( y  e.  B  |->  S ) w ) ) )
416nfcri 2622 . . . . . . . . . 10  |-  F/ x  z  e.  A
42 nffvmpt1 5872 . . . . . . . . . . . . 13  |-  F/_ x
( ( x  e.  A  |->  R ) `  z )
43 nfcv 2629 . . . . . . . . . . . . . 14  |-  F/_ x S
447, 43nfmpt 4535 . . . . . . . . . . . . 13  |-  F/_ x
( y  e.  B  |->  S )
45 nfcv 2629 . . . . . . . . . . . . 13  |-  F/_ x w
4642, 44, 45nfbr 4491 . . . . . . . . . . . 12  |-  F/ x
( ( x  e.  A  |->  R ) `  z ) ( y  e.  B  |->  S ) w
47 nfcsb1v 3451 . . . . . . . . . . . . 13  |-  F/_ x [_ z  /  x ]_ T
4847nfeq2 2646 . . . . . . . . . . . 12  |-  F/ x  w  =  [_ z  /  x ]_ T
4946, 48nfbi 1881 . . . . . . . . . . 11  |-  F/ x
( ( ( x  e.  A  |->  R ) `
 z ) ( y  e.  B  |->  S ) w  <->  w  =  [_ z  /  x ]_ T )
505, 49nfim 1867 . . . . . . . . . 10  |-  F/ x
( ph  ->  ( ( ( x  e.  A  |->  R ) `  z
) ( y  e.  B  |->  S ) w  <-> 
w  =  [_ z  /  x ]_ T ) )
5141, 50nfim 1867 . . . . . . . . 9  |-  F/ x
( z  e.  A  ->  ( ph  ->  (
( ( x  e.  A  |->  R ) `  z ) ( y  e.  B  |->  S ) w  <->  w  =  [_ z  /  x ]_ T ) ) )
52 eleq1 2539 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
53 fveq2 5864 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  (
( x  e.  A  |->  R ) `  x
)  =  ( ( x  e.  A  |->  R ) `  z ) )
5453breq1d 4457 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
( ( x  e.  A  |->  R ) `  x ) ( y  e.  B  |->  S ) w  <->  ( ( x  e.  A  |->  R ) `
 z ) ( y  e.  B  |->  S ) w ) )
55 csbeq1a 3444 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  T  =  [_ z  /  x ]_ T )
5655eqeq2d 2481 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
w  =  T  <->  w  =  [_ z  /  x ]_ T ) )
5754, 56bibi12d 321 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
( ( ( x  e.  A  |->  R ) `
 x ) ( y  e.  B  |->  S ) w  <->  w  =  T )  <->  ( (
( x  e.  A  |->  R ) `  z
) ( y  e.  B  |->  S ) w  <-> 
w  =  [_ z  /  x ]_ T ) ) )
5857imbi2d 316 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( ph  ->  ( ( ( x  e.  A  |->  R ) `  x
) ( y  e.  B  |->  S ) w  <-> 
w  =  T ) )  <->  ( ph  ->  ( ( ( x  e.  A  |->  R ) `  z ) ( y  e.  B  |->  S ) w  <->  w  =  [_ z  /  x ]_ T ) ) ) )
5952, 58imbi12d 320 . . . . . . . . 9  |-  ( x  =  z  ->  (
( x  e.  A  ->  ( ph  ->  (
( ( x  e.  A  |->  R ) `  x ) ( y  e.  B  |->  S ) w  <->  w  =  T
) ) )  <->  ( z  e.  A  ->  ( ph  ->  ( ( ( x  e.  A  |->  R ) `
 z ) ( y  e.  B  |->  S ) w  <->  w  =  [_ z  /  x ]_ T ) ) ) ) )
60 vex 3116 . . . . . . . . . . . 12  |-  w  e. 
_V
61 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( y  =  R  /\  u  =  w )  ->  y  =  R )
6261eleq1d 2536 . . . . . . . . . . . . . 14  |-  ( ( y  =  R  /\  u  =  w )  ->  ( y  e.  B  <->  R  e.  B ) )
63 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( y  =  R  /\  u  =  w )  ->  u  =  w )
64 fmptcof2.7 . . . . . . . . . . . . . . . 16  |-  ( y  =  R  ->  S  =  T )
6564adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( y  =  R  /\  u  =  w )  ->  S  =  T )
6663, 65eqeq12d 2489 . . . . . . . . . . . . . 14  |-  ( ( y  =  R  /\  u  =  w )  ->  ( u  =  S  <-> 
w  =  T ) )
6762, 66anbi12d 710 . . . . . . . . . . . . 13  |-  ( ( y  =  R  /\  u  =  w )  ->  ( ( y  e.  B  /\  u  =  S )  <->  ( R  e.  B  /\  w  =  T ) ) )
68 df-mpt 4507 . . . . . . . . . . . . 13  |-  ( y  e.  B  |->  S )  =  { <. y ,  u >.  |  (
y  e.  B  /\  u  =  S ) }
6967, 68brabga 4761 . . . . . . . . . . . 12  |-  ( ( R  e.  B  /\  w  e.  _V )  ->  ( R ( y  e.  B  |->  S ) w  <->  ( R  e.  B  /\  w  =  T ) ) )
709, 60, 69sylancl 662 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( R ( y  e.  B  |->  S ) w  <-> 
( R  e.  B  /\  w  =  T
) ) )
71 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
726fvmpt2f 27170 . . . . . . . . . . . . 13  |-  ( ( x  e.  A  /\  R  e.  B )  ->  ( ( x  e.  A  |->  R ) `  x )  =  R )
7371, 9, 72syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  R ) `  x
)  =  R )
7473breq1d 4457 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  R ) `  x ) ( y  e.  B  |->  S ) w  <->  R ( y  e.  B  |->  S ) w ) )
759biantrurd 508 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  (
w  =  T  <->  ( R  e.  B  /\  w  =  T ) ) )
7670, 74, 753bitr4d 285 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( x  e.  A  |->  R ) `  x ) ( y  e.  B  |->  S ) w  <->  w  =  T
) )
7776expcom 435 . . . . . . . . 9  |-  ( x  e.  A  ->  ( ph  ->  ( ( ( x  e.  A  |->  R ) `  x ) ( y  e.  B  |->  S ) w  <->  w  =  T ) ) )
7851, 59, 77chvar 1982 . . . . . . . 8  |-  ( z  e.  A  ->  ( ph  ->  ( ( ( x  e.  A  |->  R ) `  z ) ( y  e.  B  |->  S ) w  <->  w  =  [_ z  /  x ]_ T ) ) )
7978impcom 430 . . . . . . 7  |-  ( (
ph  /\  z  e.  A )  ->  (
( ( x  e.  A  |->  R ) `  z ) ( y  e.  B  |->  S ) w  <->  w  =  [_ z  /  x ]_ T ) )
8079pm5.32da 641 . . . . . 6  |-  ( ph  ->  ( ( z  e.  A  /\  ( ( x  e.  A  |->  R ) `  z ) ( y  e.  B  |->  S ) w )  <-> 
( z  e.  A  /\  w  =  [_ z  /  x ]_ T ) ) )
8140, 80bitrd 253 . . . . 5  |-  ( ph  ->  ( ( z F ( F `  z
)  /\  ( F `  z ) G w )  <->  ( z  e.  A  /\  w  = 
[_ z  /  x ]_ T ) ) )
8229, 81syl5bb 257 . . . 4  |-  ( ph  ->  ( E. u ( u  =  ( F `
 z )  /\  ( z F u  /\  u G w ) )  <->  ( z  e.  A  /\  w  =  [_ z  /  x ]_ T ) ) )
8324, 82bitrd 253 . . 3  |-  ( ph  ->  ( E. u ( z F u  /\  u G w )  <->  ( z  e.  A  /\  w  =  [_ z  /  x ]_ T ) ) )
84 vex 3116 . . . 4  |-  z  e. 
_V
8584, 60opelco 5172 . . 3  |-  ( <.
z ,  w >.  e.  ( G  o.  F
)  <->  E. u ( z F u  /\  u G w ) )
86 df-mpt 4507 . . . . 5  |-  ( x  e.  A  |->  T )  =  { <. x ,  v >.  |  ( x  e.  A  /\  v  =  T ) }
8786eleq2i 2545 . . . 4  |-  ( <.
z ,  w >.  e.  ( x  e.  A  |->  T )  <->  <. z ,  w >.  e.  { <. x ,  v >.  |  ( x  e.  A  /\  v  =  T ) } )
8847nfeq2 2646 . . . . . 6  |-  F/ x  v  =  [_ z  /  x ]_ T
8941, 88nfan 1875 . . . . 5  |-  F/ x
( z  e.  A  /\  v  =  [_ z  /  x ]_ T )
90 nfv 1683 . . . . 5  |-  F/ v ( z  e.  A  /\  w  =  [_ z  /  x ]_ T )
9155eqeq2d 2481 . . . . . 6  |-  ( x  =  z  ->  (
v  =  T  <->  v  =  [_ z  /  x ]_ T ) )
9252, 91anbi12d 710 . . . . 5  |-  ( x  =  z  ->  (
( x  e.  A  /\  v  =  T
)  <->  ( z  e.  A  /\  v  = 
[_ z  /  x ]_ T ) ) )
93 eqeq1 2471 . . . . . 6  |-  ( v  =  w  ->  (
v  =  [_ z  /  x ]_ T  <->  w  =  [_ z  /  x ]_ T ) )
9493anbi2d 703 . . . . 5  |-  ( v  =  w  ->  (
( z  e.  A  /\  v  =  [_ z  /  x ]_ T )  <-> 
( z  e.  A  /\  w  =  [_ z  /  x ]_ T ) ) )
9589, 90, 84, 60, 92, 94opelopabf 4772 . . . 4  |-  ( <.
z ,  w >.  e. 
{ <. x ,  v
>.  |  ( x  e.  A  /\  v  =  T ) }  <->  ( z  e.  A  /\  w  =  [_ z  /  x ]_ T ) )
9687, 95bitri 249 . . 3  |-  ( <.
z ,  w >.  e.  ( x  e.  A  |->  T )  <->  ( z  e.  A  /\  w  =  [_ z  /  x ]_ T ) )
9783, 85, 963bitr4g 288 . 2  |-  ( ph  ->  ( <. z ,  w >.  e.  ( G  o.  F )  <->  <. z ,  w >.  e.  (
x  e.  A  |->  T ) ) )
981, 4, 97eqrelrdv 5097 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596   F/wnf 1599    e. wcel 1767   F/_wnfc 2615   A.wral 2814   _Vcvv 3113   [_csb 3435   <.cop 4033   class class class wbr 4447   {copab 4504    |-> cmpt 4505   dom cdm 4999    o. ccom 5003   Rel wrel 5004   Fun wfun 5580   -->wf 5582   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594
This theorem is referenced by:  esumf1o  27701
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