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Theorem updjudhcoinlf 7092
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 7091 . . . 4  |-  ( ph  ->  H : ( A B ) --> C )
5 ffn 5377 . . . 4  |-  ( H : ( A B ) --> C  ->  H  Fn  ( A B ) )
64, 5syl 14 . . 3  |-  ( ph  ->  H  Fn  ( A B ) )
7 inlresf1 7073 . . . 4  |-  (inl  |`  A ) : A -1-1-> ( A B )
8 f1fn 5435 . . . 4  |-  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  (inl  |`  A )  Fn  A
)
97, 8mp1i 10 . . 3  |-  ( ph  ->  (inl  |`  A )  Fn  A )
10 f1f 5433 . . . . 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 5386 . . . 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 5336 . . 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 1248 . 2  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  Fn  A )
16 ffn 5377 . . 3  |-  ( F : A --> C  ->  F  Fn  A )
171, 16syl 14 . 2  |-  ( ph  ->  F  Fn  A )
18 fvco2 5598 . . . 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 5551 . . . . . 6  |-  ( a  e.  A  ->  (
(inl  |`  A ) `  a )  =  (inl
`  a ) )
2120adantl 277 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (
(inl  |`  A ) `  a )  =  (inl
`  a ) )
2221fveq2d 5531 . . . 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 5527 . . . . . . . . 9  |-  ( x  =  (inl `  a
)  ->  ( 1st `  x )  =  ( 1st `  (inl `  a ) ) )
2524eqeq1d 2196 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( ( 1st `  x )  =  (/) 
<->  ( 1st `  (inl `  a ) )  =  (/) ) )
26 fveq2 5527 . . . . . . . . 9  |-  ( x  =  (inl `  a
)  ->  ( 2nd `  x )  =  ( 2nd `  (inl `  a ) ) )
2726fveq2d 5531 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( F `  ( 2nd `  x
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
2826fveq2d 5531 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( G `  ( 2nd `  x
) )  =  ( G `  ( 2nd `  (inl `  a )
) ) )
2925, 27, 28ifbieq12d 3572 . . . . . . 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 7086 . . . . . . . . 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 3553 . . . . . 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 2220 . . . . 5  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  ( F `
 ( 2nd `  (inl `  a ) ) ) )
36 djulcl 7063 . . . . . 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 7087 . . . . . . . 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 2264 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  ( 2nd `  (inl `  a
) )  e.  A
)
4338, 42ffvelcdmd 5665 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  ( 2nd `  (inl `  a )
) )  e.  C
)
4423, 35, 37, 43fvmptd 5610 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  (inl `  a
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
4522, 44eqtrd 2220 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  ( (inl  |`  A ) `  a
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
4640fveq2d 5531 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  ( 2nd `  (inl `  a )
) )  =  ( F `  a ) )
4719, 45, 463eqtrd 2224 . 2  |-  ( (
ph  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( F `  a ) )
4815, 17, 47eqfnfvd 5629 1  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158    C_ wss 3141   (/)c0 3434   ifcif 3546    |-> cmpt 4076   ran crn 4639    |` cres 4640    o. ccom 4642    Fn wfn 5223   -->wf 5224   -1-1->wf1 5225   ` cfv 5228   1stc1st 6152   2ndc2nd 6153   ⊔ cdju 7049  inlcinl 7057
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6154  df-2nd 6155  df-1o 6430  df-dju 7050  df-inl 7059  df-inr 7060
This theorem is referenced by:  updjud  7094
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