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Theorem updjudhcoinlf 7099
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 7098 . . . 4  |-  ( ph  ->  H : ( A B ) --> C )
5 ffn 5381 . . . 4  |-  ( H : ( A B ) --> C  ->  H  Fn  ( A B ) )
64, 5syl 14 . . 3  |-  ( ph  ->  H  Fn  ( A B ) )
7 inlresf1 7080 . . . 4  |-  (inl  |`  A ) : A -1-1-> ( A B )
8 f1fn 5439 . . . 4  |-  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  (inl  |`  A )  Fn  A
)
97, 8mp1i 10 . . 3  |-  ( ph  ->  (inl  |`  A )  Fn  A )
10 f1f 5437 . . . . 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 5390 . . . 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 5340 . . 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 1249 . 2  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  Fn  A )
16 ffn 5381 . . 3  |-  ( F : A --> C  ->  F  Fn  A )
171, 16syl 14 . 2  |-  ( ph  ->  F  Fn  A )
18 fvco2 5602 . . . 4  |-  ( ( (inl  |`  A )  Fn  A  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( H `  ( (inl  |`  A ) `  a
) ) )
199, 18sylan 283 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( H `  ( (inl  |`  A ) `  a
) ) )
20 fvres 5555 . . . . . 6  |-  ( a  e.  A  ->  (
(inl  |`  A ) `  a )  =  (inl
`  a ) )
2120adantl 277 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (
(inl  |`  A ) `  a )  =  (inl
`  a ) )
2221fveq2d 5535 . . . 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 5531 . . . . . . . . 9  |-  ( x  =  (inl `  a
)  ->  ( 1st `  x )  =  ( 1st `  (inl `  a ) ) )
2524eqeq1d 2198 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( ( 1st `  x )  =  (/) 
<->  ( 1st `  (inl `  a ) )  =  (/) ) )
26 fveq2 5531 . . . . . . . . 9  |-  ( x  =  (inl `  a
)  ->  ( 2nd `  x )  =  ( 2nd `  (inl `  a ) ) )
2726fveq2d 5535 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( F `  ( 2nd `  x
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
2826fveq2d 5535 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( G `  ( 2nd `  x
) )  =  ( G `  ( 2nd `  (inl `  a )
) ) )
2925, 27, 28ifbieq12d 3575 . . . . . . 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 277 . . . . . 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 7093 . . . . . . . . 9  |-  ( a  e.  A  ->  ( 1st `  (inl `  a
) )  =  (/) )
3231adantl 277 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( 1st `  (inl `  a
) )  =  (/) )
3332adantr 276 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  ( 1st `  (inl `  a )
)  =  (/) )
3433iftrued 3556 . . . . . 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 2222 . . . . 5  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  ( F `
 ( 2nd `  (inl `  a ) ) ) )
36 djulcl 7070 . . . . . 6  |-  ( a  e.  A  ->  (inl `  a )  e.  ( A B ) )
3736adantl 277 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (inl `  a )  e.  ( A B ) )
381adantr 276 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  F : A --> C )
39 2ndinl 7094 . . . . . . . 8  |-  ( a  e.  A  ->  ( 2nd `  (inl `  a
) )  =  a )
4039adantl 277 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( 2nd `  (inl `  a
) )  =  a )
41 simpr 110 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  A )
4240, 41eqeltrd 2266 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  ( 2nd `  (inl `  a
) )  e.  A
)
4338, 42ffvelcdmd 5669 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  ( 2nd `  (inl `  a )
) )  e.  C
)
4423, 35, 37, 43fvmptd 5614 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  (inl `  a
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
4522, 44eqtrd 2222 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  ( (inl  |`  A ) `  a
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
4640fveq2d 5535 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  ( 2nd `  (inl `  a )
) )  =  ( F `  a ) )
4719, 45, 463eqtrd 2226 . 2  |-  ( (
ph  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( F `  a ) )
4815, 17, 47eqfnfvd 5633 1  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160    C_ wss 3144   (/)c0 3437   ifcif 3549    |-> cmpt 4079   ran crn 4642    |` cres 4643    o. ccom 4645    Fn wfn 5227   -->wf 5228   -1-1->wf1 5229   ` cfv 5232   1stc1st 6158   2ndc2nd 6159   ⊔ cdju 7056  inlcinl 7064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-1st 6160  df-2nd 6161  df-1o 6436  df-dju 7057  df-inl 7066  df-inr 7067
This theorem is referenced by:  updjud  7101
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