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Theorem updjudhcoinrg 6932
Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)
Hypotheses
Ref Expression
updjud.f  |-  ( ph  ->  F : A --> C )
updjud.g  |-  ( ph  ->  G : B --> C )
updjudhf.h  |-  H  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )
Assertion
Ref Expression
updjudhcoinrg  |-  ( ph  ->  ( H  o.  (inr  |`  B ) )  =  G )
Distinct variable groups:    x, A    x, B    x, C    ph, x    x, F    x, G
Allowed substitution hint:    H( x)

Proof of Theorem updjudhcoinrg
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 updjud.f . . . . 5  |-  ( ph  ->  F : A --> C )
2 updjud.g . . . . 5  |-  ( ph  ->  G : B --> C )
3 updjudhf.h . . . . 5  |-  H  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )
41, 2, 3updjudhf 6930 . . . 4  |-  ( ph  ->  H : ( A B ) --> C )
5 ffn 5240 . . . 4  |-  ( H : ( A B ) --> C  ->  H  Fn  ( A B ) )
64, 5syl 14 . . 3  |-  ( ph  ->  H  Fn  ( A B ) )
7 inrresf1 6913 . . . 4  |-  (inr  |`  B ) : B -1-1-> ( A B )
8 f1fn 5298 . . . 4  |-  ( (inr  |`  B ) : B -1-1-> ( A B )  ->  (inr  |`  B )  Fn  B
)
97, 8mp1i 10 . . 3  |-  ( ph  ->  (inr  |`  B )  Fn  B )
10 f1f 5296 . . . . 5  |-  ( (inr  |`  B ) : B -1-1-> ( A B )  ->  (inr  |`  B ) : B --> ( A B ) )
117, 10ax-mp 5 . . . 4  |-  (inr  |`  B ) : B --> ( A B )
12 frn 5249 . . . 4  |-  ( (inr  |`  B ) : B --> ( A B )  ->  ran  (inr  |`  B )  C_  ( A B ) )
1311, 12mp1i 10 . . 3  |-  ( ph  ->  ran  (inr  |`  B ) 
C_  ( A B ) )
14 fnco 5199 . . 3  |-  ( ( H  Fn  ( A B )  /\  (inr  |`  B )  Fn  B  /\  ran  (inr  |`  B ) 
C_  ( A B ) )  ->  ( H  o.  (inr  |`  B ) )  Fn  B )
156, 9, 13, 14syl3anc 1199 . 2  |-  ( ph  ->  ( H  o.  (inr  |`  B ) )  Fn  B )
16 ffn 5240 . . 3  |-  ( G : B --> C  ->  G  Fn  B )
172, 16syl 14 . 2  |-  ( ph  ->  G  Fn  B )
18 fvco2 5456 . . . 4  |-  ( ( (inr  |`  B )  Fn  B  /\  b  e.  B )  ->  (
( H  o.  (inr  |`  B ) ) `  b )  =  ( H `  ( (inr  |`  B ) `  b
) ) )
199, 18sylan 279 . . 3  |-  ( (
ph  /\  b  e.  B )  ->  (
( H  o.  (inr  |`  B ) ) `  b )  =  ( H `  ( (inr  |`  B ) `  b
) ) )
20 fvres 5411 . . . . . 6  |-  ( b  e.  B  ->  (
(inr  |`  B ) `  b )  =  (inr
`  b ) )
2120adantl 273 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  (
(inr  |`  B ) `  b )  =  (inr
`  b ) )
2221fveq2d 5391 . . . 4  |-  ( (
ph  /\  b  e.  B )  ->  ( H `  ( (inr  |`  B ) `  b
) )  =  ( H `  (inr `  b ) ) )
233a1i 9 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  H  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) )
24 fveq2 5387 . . . . . . . . 9  |-  ( x  =  (inr `  b
)  ->  ( 1st `  x )  =  ( 1st `  (inr `  b ) ) )
2524eqeq1d 2124 . . . . . . . 8  |-  ( x  =  (inr `  b
)  ->  ( ( 1st `  x )  =  (/) 
<->  ( 1st `  (inr `  b ) )  =  (/) ) )
26 fveq2 5387 . . . . . . . . 9  |-  ( x  =  (inr `  b
)  ->  ( 2nd `  x )  =  ( 2nd `  (inr `  b ) ) )
2726fveq2d 5391 . . . . . . . 8  |-  ( x  =  (inr `  b
)  ->  ( F `  ( 2nd `  x
) )  =  ( F `  ( 2nd `  (inr `  b )
) ) )
2826fveq2d 5391 . . . . . . . 8  |-  ( x  =  (inr `  b
)  ->  ( G `  ( 2nd `  x
) )  =  ( G `  ( 2nd `  (inr `  b )
) ) )
2925, 27, 28ifbieq12d 3466 . . . . . . 7  |-  ( x  =  (inr `  b
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  if ( ( 1st `  (inr `  b ) )  =  (/) ,  ( F `  ( 2nd `  (inr `  b ) ) ) ,  ( G `  ( 2nd `  (inr `  b ) ) ) ) )
3029adantl 273 . . . . . 6  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  (inr `  b )
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  if ( ( 1st `  (inr `  b ) )  =  (/) ,  ( F `  ( 2nd `  (inr `  b ) ) ) ,  ( G `  ( 2nd `  (inr `  b ) ) ) ) )
31 1stinr 6927 . . . . . . . . . 10  |-  ( b  e.  B  ->  ( 1st `  (inr `  b
) )  =  1o )
32 1n0 6295 . . . . . . . . . . . 12  |-  1o  =/=  (/)
3332neii 2285 . . . . . . . . . . 11  |-  -.  1o  =  (/)
34 eqeq1 2122 . . . . . . . . . . 11  |-  ( ( 1st `  (inr `  b ) )  =  1o  ->  ( ( 1st `  (inr `  b
) )  =  (/)  <->  1o  =  (/) ) )
3533, 34mtbiri 647 . . . . . . . . . 10  |-  ( ( 1st `  (inr `  b ) )  =  1o  ->  -.  ( 1st `  (inr `  b
) )  =  (/) )
3631, 35syl 14 . . . . . . . . 9  |-  ( b  e.  B  ->  -.  ( 1st `  (inr `  b ) )  =  (/) )
3736adantl 273 . . . . . . . 8  |-  ( (
ph  /\  b  e.  B )  ->  -.  ( 1st `  (inr `  b ) )  =  (/) )
3837adantr 272 . . . . . . 7  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  (inr `  b )
)  ->  -.  ( 1st `  (inr `  b
) )  =  (/) )
3938iffalsed 3452 . . . . . 6  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  (inr `  b )
)  ->  if (
( 1st `  (inr `  b ) )  =  (/) ,  ( F `  ( 2nd `  (inr `  b ) ) ) ,  ( G `  ( 2nd `  (inr `  b ) ) ) )  =  ( G `
 ( 2nd `  (inr `  b ) ) ) )
4030, 39eqtrd 2148 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  (inr `  b )
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  ( G `
 ( 2nd `  (inr `  b ) ) ) )
41 djurcl 6903 . . . . . 6  |-  ( b  e.  B  ->  (inr `  b )  e.  ( A B ) )
4241adantl 273 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  (inr `  b )  e.  ( A B ) )
432adantr 272 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  G : B --> C )
44 2ndinr 6928 . . . . . . . 8  |-  ( b  e.  B  ->  ( 2nd `  (inr `  b
) )  =  b )
4544adantl 273 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  ( 2nd `  (inr `  b
) )  =  b )
46 simpr 109 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  B )
4745, 46eqeltrd 2192 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  ( 2nd `  (inr `  b
) )  e.  B
)
4843, 47ffvelrnd 5522 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  ( G `  ( 2nd `  (inr `  b )
) )  e.  C
)
4923, 40, 42, 48fvmptd 5468 . . . 4  |-  ( (
ph  /\  b  e.  B )  ->  ( H `  (inr `  b
) )  =  ( G `  ( 2nd `  (inr `  b )
) ) )
5022, 49eqtrd 2148 . . 3  |-  ( (
ph  /\  b  e.  B )  ->  ( H `  ( (inr  |`  B ) `  b
) )  =  ( G `  ( 2nd `  (inr `  b )
) ) )
5145fveq2d 5391 . . 3  |-  ( (
ph  /\  b  e.  B )  ->  ( G `  ( 2nd `  (inr `  b )
) )  =  ( G `  b ) )
5219, 50, 513eqtrd 2152 . 2  |-  ( (
ph  /\  b  e.  B )  ->  (
( H  o.  (inr  |`  B ) ) `  b )  =  ( G `  b ) )
5315, 17, 52eqfnfvd 5487 1  |-  ( ph  ->  ( H  o.  (inr  |`  B ) )  =  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463    C_ wss 3039   (/)c0 3331   ifcif 3442    |-> cmpt 3957   ran crn 4508    |` cres 4509    o. ccom 4511    Fn wfn 5086   -->wf 5087   -1-1->wf1 5088   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   1oc1o 6272   ⊔ cdju 6888  inrcinr 6897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by:  updjud  6933
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