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Theorem updjudhcoinrg 6934
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 6932 . . . 4  |-  ( ph  ->  H : ( A B ) --> C )
5 ffn 5242 . . . 4  |-  ( H : ( A B ) --> C  ->  H  Fn  ( A B ) )
64, 5syl 14 . . 3  |-  ( ph  ->  H  Fn  ( A B ) )
7 inrresf1 6915 . . . 4  |-  (inr  |`  B ) : B -1-1-> ( A B )
8 f1fn 5300 . . . 4  |-  ( (inr  |`  B ) : B -1-1-> ( A B )  ->  (inr  |`  B )  Fn  B
)
97, 8mp1i 10 . . 3  |-  ( ph  ->  (inr  |`  B )  Fn  B )
10 f1f 5298 . . . . 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 5251 . . . 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 5201 . . 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 1201 . 2  |-  ( ph  ->  ( H  o.  (inr  |`  B ) )  Fn  B )
16 ffn 5242 . . 3  |-  ( G : B --> C  ->  G  Fn  B )
172, 16syl 14 . 2  |-  ( ph  ->  G  Fn  B )
18 fvco2 5458 . . . 4  |-  ( ( (inr  |`  B )  Fn  B  /\  b  e.  B )  ->  (
( H  o.  (inr  |`  B ) ) `  b )  =  ( H `  ( (inr  |`  B ) `  b
) ) )
199, 18sylan 281 . . 3  |-  ( (
ph  /\  b  e.  B )  ->  (
( H  o.  (inr  |`  B ) ) `  b )  =  ( H `  ( (inr  |`  B ) `  b
) ) )
20 fvres 5413 . . . . . 6  |-  ( b  e.  B  ->  (
(inr  |`  B ) `  b )  =  (inr
`  b ) )
2120adantl 275 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  (
(inr  |`  B ) `  b )  =  (inr
`  b ) )
2221fveq2d 5393 . . . 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 5389 . . . . . . . . 9  |-  ( x  =  (inr `  b
)  ->  ( 1st `  x )  =  ( 1st `  (inr `  b ) ) )
2524eqeq1d 2126 . . . . . . . 8  |-  ( x  =  (inr `  b
)  ->  ( ( 1st `  x )  =  (/) 
<->  ( 1st `  (inr `  b ) )  =  (/) ) )
26 fveq2 5389 . . . . . . . . 9  |-  ( x  =  (inr `  b
)  ->  ( 2nd `  x )  =  ( 2nd `  (inr `  b ) ) )
2726fveq2d 5393 . . . . . . . 8  |-  ( x  =  (inr `  b
)  ->  ( F `  ( 2nd `  x
) )  =  ( F `  ( 2nd `  (inr `  b )
) ) )
2826fveq2d 5393 . . . . . . . 8  |-  ( x  =  (inr `  b
)  ->  ( G `  ( 2nd `  x
) )  =  ( G `  ( 2nd `  (inr `  b )
) ) )
2925, 27, 28ifbieq12d 3468 . . . . . . 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 275 . . . . . 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 6929 . . . . . . . . . 10  |-  ( b  e.  B  ->  ( 1st `  (inr `  b
) )  =  1o )
32 1n0 6297 . . . . . . . . . . . 12  |-  1o  =/=  (/)
3332neii 2287 . . . . . . . . . . 11  |-  -.  1o  =  (/)
34 eqeq1 2124 . . . . . . . . . . 11  |-  ( ( 1st `  (inr `  b ) )  =  1o  ->  ( ( 1st `  (inr `  b
) )  =  (/)  <->  1o  =  (/) ) )
3533, 34mtbiri 649 . . . . . . . . . 10  |-  ( ( 1st `  (inr `  b ) )  =  1o  ->  -.  ( 1st `  (inr `  b
) )  =  (/) )
3631, 35syl 14 . . . . . . . . 9  |-  ( b  e.  B  ->  -.  ( 1st `  (inr `  b ) )  =  (/) )
3736adantl 275 . . . . . . . 8  |-  ( (
ph  /\  b  e.  B )  ->  -.  ( 1st `  (inr `  b ) )  =  (/) )
3837adantr 274 . . . . . . 7  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  (inr `  b )
)  ->  -.  ( 1st `  (inr `  b
) )  =  (/) )
3938iffalsed 3454 . . . . . 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 2150 . . . . 5  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  (inr `  b )
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  ( G `
 ( 2nd `  (inr `  b ) ) ) )
41 djurcl 6905 . . . . . 6  |-  ( b  e.  B  ->  (inr `  b )  e.  ( A B ) )
4241adantl 275 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  (inr `  b )  e.  ( A B ) )
432adantr 274 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  G : B --> C )
44 2ndinr 6930 . . . . . . . 8  |-  ( b  e.  B  ->  ( 2nd `  (inr `  b
) )  =  b )
4544adantl 275 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  ( 2nd `  (inr `  b
) )  =  b )
46 simpr 109 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  B )
4745, 46eqeltrd 2194 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  ( 2nd `  (inr `  b
) )  e.  B
)
4843, 47ffvelrnd 5524 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  ( G `  ( 2nd `  (inr `  b )
) )  e.  C
)
4923, 40, 42, 48fvmptd 5470 . . . 4  |-  ( (
ph  /\  b  e.  B )  ->  ( H `  (inr `  b
) )  =  ( G `  ( 2nd `  (inr `  b )
) ) )
5022, 49eqtrd 2150 . . 3  |-  ( (
ph  /\  b  e.  B )  ->  ( H `  ( (inr  |`  B ) `  b
) )  =  ( G `  ( 2nd `  (inr `  b )
) ) )
5145fveq2d 5393 . . 3  |-  ( (
ph  /\  b  e.  B )  ->  ( G `  ( 2nd `  (inr `  b )
) )  =  ( G `  b ) )
5219, 50, 513eqtrd 2154 . 2  |-  ( (
ph  /\  b  e.  B )  ->  (
( H  o.  (inr  |`  B ) ) `  b )  =  ( G `  b ) )
5315, 17, 52eqfnfvd 5489 1  |-  ( ph  ->  ( H  o.  (inr  |`  B ) )  =  G )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465    C_ wss 3041   (/)c0 3333   ifcif 3444    |-> cmpt 3959   ran crn 4510    |` cres 4511    o. ccom 4513    Fn wfn 5088   -->wf 5089   -1-1->wf1 5090   ` cfv 5093   1stc1st 6004   2ndc2nd 6005   1oc1o 6274   ⊔ cdju 6890  inrcinr 6899
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901
This theorem is referenced by:  updjud  6935
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