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Theorem updjudhcoinlf 6769
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 6768 . . . 4  |-  ( ph  ->  H : ( A B ) --> C )
5 ffn 5161 . . . 4  |-  ( H : ( A B ) --> C  ->  H  Fn  ( A B ) )
64, 5syl 14 . . 3  |-  ( ph  ->  H  Fn  ( A B ) )
7 inlresf1 6751 . . . 4  |-  (inl  |`  A ) : A -1-1-> ( A B )
8 f1fn 5218 . . . 4  |-  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  (inl  |`  A )  Fn  A
)
97, 8mp1i 10 . . 3  |-  ( ph  ->  (inl  |`  A )  Fn  A )
10 f1f 5216 . . . . 5  |-  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  (inl  |`  A ) : A --> ( A B ) )
117, 10ax-mp 7 . . . 4  |-  (inl  |`  A ) : A --> ( A B )
12 frn 5169 . . . 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 5122 . . 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 1174 . 2  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  Fn  A )
16 ffn 5161 . . 3  |-  ( F : A --> C  ->  F  Fn  A )
171, 16syl 14 . 2  |-  ( ph  ->  F  Fn  A )
18 fvco2 5373 . . . 4  |-  ( ( (inl  |`  A )  Fn  A  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( H `  ( (inl  |`  A ) `  a
) ) )
199, 18sylan 277 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( H `  ( (inl  |`  A ) `  a
) ) )
20 fvres 5329 . . . . . 6  |-  ( a  e.  A  ->  (
(inl  |`  A ) `  a )  =  (inl
`  a ) )
2120adantl 271 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (
(inl  |`  A ) `  a )  =  (inl
`  a ) )
2221fveq2d 5309 . . . 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 5305 . . . . . . . . 9  |-  ( x  =  (inl `  a
)  ->  ( 1st `  x )  =  ( 1st `  (inl `  a ) ) )
2524eqeq1d 2096 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( ( 1st `  x )  =  (/) 
<->  ( 1st `  (inl `  a ) )  =  (/) ) )
26 fveq2 5305 . . . . . . . . 9  |-  ( x  =  (inl `  a
)  ->  ( 2nd `  x )  =  ( 2nd `  (inl `  a ) ) )
2726fveq2d 5309 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( F `  ( 2nd `  x
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
2826fveq2d 5309 . . . . . . . 8  |-  ( x  =  (inl `  a
)  ->  ( G `  ( 2nd `  x
) )  =  ( G `  ( 2nd `  (inl `  a )
) ) )
2925, 27, 28ifbieq12d 3417 . . . . . . 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 271 . . . . . 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 6763 . . . . . . . . 9  |-  ( a  e.  A  ->  ( 1st `  (inl `  a
) )  =  (/) )
3231adantl 271 . . . . . . . 8  |-  ( (
ph  /\  a  e.  A )  ->  ( 1st `  (inl `  a
) )  =  (/) )
3332adantr 270 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  ( 1st `  (inl `  a )
)  =  (/) )
3433iftrued 3400 . . . . . 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 2120 . . . . 5  |-  ( ( ( ph  /\  a  e.  A )  /\  x  =  (inl `  a )
)  ->  if (
( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) )  =  ( F `
 ( 2nd `  (inl `  a ) ) ) )
36 djulcl 6741 . . . . . 6  |-  ( a  e.  A  ->  (inl `  a )  e.  ( A B ) )
3736adantl 271 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (inl `  a )  e.  ( A B ) )
381adantr 270 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  F : A --> C )
39 2ndinl 6764 . . . . . . . 8  |-  ( a  e.  A  ->  ( 2nd `  (inl `  a
) )  =  a )
4039adantl 271 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( 2nd `  (inl `  a
) )  =  a )
41 simpr 108 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  a  e.  A )
4240, 41eqeltrd 2164 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  ( 2nd `  (inl `  a
) )  e.  A
)
4338, 42ffvelrnd 5435 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  ( 2nd `  (inl `  a )
) )  e.  C
)
4423, 35, 37, 43fvmptd 5385 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  (inl `  a
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
4522, 44eqtrd 2120 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( H `  ( (inl  |`  A ) `  a
) )  =  ( F `  ( 2nd `  (inl `  a )
) ) )
4640fveq2d 5309 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  ( 2nd `  (inl `  a )
) )  =  ( F `  a ) )
4719, 45, 463eqtrd 2124 . 2  |-  ( (
ph  /\  a  e.  A )  ->  (
( H  o.  (inl  |`  A ) ) `  a )  =  ( F `  a ) )
4815, 17, 47eqfnfvd 5400 1  |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    C_ wss 2999   (/)c0 3286   ifcif 3393    |-> cmpt 3899   ran crn 4439    |` cres 4440    o. ccom 4442    Fn wfn 5010   -->wf 5011   -1-1->wf1 5012   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   ⊔ cdju 6728  inlcinl 6735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-dju 6729  df-inl 6737  df-inr 6738
This theorem is referenced by:  updjud  6771
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