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Theorem updjudhcoinlf 6958
Description: The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first 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
updjudhcoinlf  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  =  F )
Distinct variable groups:    x, A    x, B    x, C    ph, x    x, F
Allowed substitution hints:    G( x)    H( x)

Proof of Theorem updjudhcoinlf
Dummy variable  a 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 6957 . . . 4  |-  ( ph  ->  H : ( A B ) --> C )
5 ffn 5267 . . . 4  |-  ( H : ( A B ) --> C  ->  H  Fn  ( A B ) )
64, 5syl 14 . . 3  |-  ( ph  ->  H  Fn  ( A B ) )
7 inlresf1 6939 . . . 4  |-  (inl  |`  A ) : A -1-1-> ( A B )
8 f1fn 5325 . . . 4  |-  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  (inl  |`  A )  Fn  A
)
97, 8mp1i 10 . . 3  |-  ( ph  ->  (inl  |`  A )  Fn  A )
10 f1f 5323 . . . . 5  |-  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  (inl  |`  A ) : A --> ( A B ) )
117, 10ax-mp 5 . . . 4  |-  (inl  |`  A ) : A --> ( A B )
12 frn 5276 . . . 4  |-  ( (inl  |`  A ) : A --> ( A B )  ->  ran  (inl  |`  A )  C_  ( A B ) )
1311, 12mp1i 10 . . 3  |-  ( ph  ->  ran  (inl  |`  A ) 
C_  ( A B ) )
14 fnco 5226 . . 3  |-  ( ( H  Fn  ( A B )  /\  (inl  |`  A )  Fn  A  /\  ran  (inl  |`  A ) 
C_  ( A B ) )  ->  ( H  o.  (inl  |`  A ) )  Fn  A )
156, 9, 13, 14syl3anc 1216 . 2  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  Fn  A )
16 ffn 5267 . . 3  |-  ( F : A --> C  ->  F  Fn  A )
171, 16syl 14 . 2  |-  ( ph  ->  F  Fn  A )
18 fvco2 5483 . . . 4  |-  ( ( (inl  |`  A )  Fn  A  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( H `  ( (inl  |`  A ) `  a
) ) )
199, 18sylan 281 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( H `  ( (inl  |`  A ) `  a
) ) )
20 fvres 5438 . . . . . 6  |-  ( a  e.  A  ->  (
(inl  |`  A ) `  a )  =  (inl
`  a ) )
2120adantl 275 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (
(inl  |`  A ) `  a )  =  (inl
`  a ) )
2221fveq2d 5418 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  ( (inl  |`  A ) `  a
) )  =  ( H `  (inl `  a ) ) )
233a1i 9 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  H  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x )  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) ) )
24 fveq2 5414 . . . . . . . . 9  |-  ( x  =  (inl `  a
)  ->  ( 1st `  x )  =  ( 1st `  (inl `  a ) ) )
2524eqeq1d 2146 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( ( 1st `  x )  =  (/) 
<->  ( 1st `  (inl `  a ) )  =  (/) ) )
26 fveq2 5414 . . . . . . . . 9  |-  ( x  =  (inl `  a
)  ->  ( 2nd `  x )  =  ( 2nd `  (inl `  a ) ) )
2726fveq2d 5418 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( F `  ( 2nd `  x
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
2826fveq2d 5418 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( G `  ( 2nd `  x
) )  =  ( G `  ( 2nd `  (inl `  a )
) ) )
2925, 27, 28ifbieq12d 3493 . . . . . . 7  |-  ( x  =  (inl `  a
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  if ( ( 1st `  (inl `  a ) )  =  (/) ,  ( F `  ( 2nd `  (inl `  a ) ) ) ,  ( G `  ( 2nd `  (inl `  a ) ) ) ) )
3029adantl 275 . . . . . 6  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  if ( ( 1st `  (inl `  a ) )  =  (/) ,  ( F `  ( 2nd `  (inl `  a ) ) ) ,  ( G `  ( 2nd `  (inl `  a ) ) ) ) )
31 1stinl 6952 . . . . . . . . 9  |-  ( a  e.  A  ->  ( 1st `  (inl `  a
) )  =  (/) )
3231adantl 275 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( 1st `  (inl `  a
) )  =  (/) )
3332adantr 274 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  ( 1st `  (inl `  a )
)  =  (/) )
3433iftrued 3476 . . . . . 6  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  if (
( 1st `  (inl `  a ) )  =  (/) ,  ( F `  ( 2nd `  (inl `  a ) ) ) ,  ( G `  ( 2nd `  (inl `  a ) ) ) )  =  ( F `
 ( 2nd `  (inl `  a ) ) ) )
3530, 34eqtrd 2170 . . . . 5  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  ( F `
 ( 2nd `  (inl `  a ) ) ) )
36 djulcl 6929 . . . . . 6  |-  ( a  e.  A  ->  (inl `  a )  e.  ( A B ) )
3736adantl 275 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (inl `  a )  e.  ( A B ) )
381adantr 274 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  F : A --> C )
39 2ndinl 6953 . . . . . . . 8  |-  ( a  e.  A  ->  ( 2nd `  (inl `  a
) )  =  a )
4039adantl 275 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( 2nd `  (inl `  a
) )  =  a )
41 simpr 109 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  A )
4240, 41eqeltrd 2214 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  ( 2nd `  (inl `  a
) )  e.  A
)
4338, 42ffvelrnd 5549 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  ( 2nd `  (inl `  a )
) )  e.  C
)
4423, 35, 37, 43fvmptd 5495 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  (inl `  a
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
4522, 44eqtrd 2170 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  ( (inl  |`  A ) `  a
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
4640fveq2d 5418 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  ( 2nd `  (inl `  a )
) )  =  ( F `  a ) )
4719, 45, 463eqtrd 2174 . 2  |-  ( (
ph  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( F `  a ) )
4815, 17, 47eqfnfvd 5514 1  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    C_ wss 3066   (/)c0 3358   ifcif 3469    |-> cmpt 3984   ran crn 4535    |` cres 4536    o. ccom 4538    Fn wfn 5113   -->wf 5114   -1-1->wf1 5115   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   ⊔ cdju 6915  inlcinl 6923
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dju 6916  df-inl 6925  df-inr 6926
This theorem is referenced by:  updjud  6960
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