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Theorem updjudhf 6770
Description: The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-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
updjudhf  |-  ( ph  ->  H : ( A B ) --> C )
Distinct variable groups:    x, A    x, B    x, C    ph, x
Allowed substitution hints:    F( x)    G( x)    H( x)

Proof of Theorem updjudhf
StepHypRef Expression
1 eldju2ndl 6763 . . . . . 6  |-  ( ( x  e.  ( A B )  /\  ( 1st `  x )  =  (/) )  ->  ( 2nd `  x )  e.  A
)
21ex 113 . . . . 5  |-  ( x  e.  ( A B )  ->  ( ( 1st `  x )  =  (/)  ->  ( 2nd `  x
)  e.  A ) )
3 updjud.f . . . . . 6  |-  ( ph  ->  F : A --> C )
4 ffvelrn 5432 . . . . . . 7  |-  ( ( F : A --> C  /\  ( 2nd `  x )  e.  A )  -> 
( F `  ( 2nd `  x ) )  e.  C )
54ex 113 . . . . . 6  |-  ( F : A --> C  -> 
( ( 2nd `  x
)  e.  A  -> 
( F `  ( 2nd `  x ) )  e.  C ) )
63, 5syl 14 . . . . 5  |-  ( ph  ->  ( ( 2nd `  x
)  e.  A  -> 
( F `  ( 2nd `  x ) )  e.  C ) )
72, 6sylan9r 402 . . . 4  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  ( ( 1st `  x )  =  (/)  ->  ( F `  ( 2nd `  x ) )  e.  C ) )
87imp 122 . . 3  |-  ( ( ( ph  /\  x  e.  ( A B )
)  /\  ( 1st `  x )  =  (/) )  ->  ( F `  ( 2nd `  x ) )  e.  C )
9 df-ne 2256 . . . . 5  |-  ( ( 1st `  x )  =/=  (/)  <->  -.  ( 1st `  x )  =  (/) )
10 eldju2ndr 6764 . . . . . . 7  |-  ( ( x  e.  ( A B )  /\  ( 1st `  x )  =/=  (/) )  ->  ( 2nd `  x )  e.  B
)
1110ex 113 . . . . . 6  |-  ( x  e.  ( A B )  ->  ( ( 1st `  x )  =/=  (/)  ->  ( 2nd `  x )  e.  B ) )
12 updjud.g . . . . . . 7  |-  ( ph  ->  G : B --> C )
13 ffvelrn 5432 . . . . . . . 8  |-  ( ( G : B --> C  /\  ( 2nd `  x )  e.  B )  -> 
( G `  ( 2nd `  x ) )  e.  C )
1413ex 113 . . . . . . 7  |-  ( G : B --> C  -> 
( ( 2nd `  x
)  e.  B  -> 
( G `  ( 2nd `  x ) )  e.  C ) )
1512, 14syl 14 . . . . . 6  |-  ( ph  ->  ( ( 2nd `  x
)  e.  B  -> 
( G `  ( 2nd `  x ) )  e.  C ) )
1611, 15sylan9r 402 . . . . 5  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  ( ( 1st `  x )  =/=  (/)  ->  ( G `  ( 2nd `  x ) )  e.  C ) )
179, 16syl5bir 151 . . . 4  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  ( -.  ( 1st `  x )  =  (/)  ->  ( G `  ( 2nd `  x ) )  e.  C ) )
1817imp 122 . . 3  |-  ( ( ( ph  /\  x  e.  ( A B )
)  /\  -.  ( 1st `  x )  =  (/) )  ->  ( G `
 ( 2nd `  x
) )  e.  C
)
19 eldju1st 6762 . . . . . 6  |-  ( x  e.  ( A B )  ->  ( ( 1st `  x )  =  (/)  \/  ( 1st `  x
)  =  1o ) )
20 1n0 6197 . . . . . . . 8  |-  1o  =/=  (/)
21 neeq1 2268 . . . . . . . 8  |-  ( ( 1st `  x )  =  1o  ->  (
( 1st `  x
)  =/=  (/)  <->  1o  =/=  (/) ) )
2220, 21mpbiri 166 . . . . . . 7  |-  ( ( 1st `  x )  =  1o  ->  ( 1st `  x )  =/=  (/) )
2322orim2i 713 . . . . . 6  |-  ( ( ( 1st `  x
)  =  (/)  \/  ( 1st `  x )  =  1o )  ->  (
( 1st `  x
)  =  (/)  \/  ( 1st `  x )  =/=  (/) ) )
2419, 23syl 14 . . . . 5  |-  ( x  e.  ( A B )  ->  ( ( 1st `  x )  =  (/)  \/  ( 1st `  x
)  =/=  (/) ) )
2524adantl 271 . . . 4  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  ( ( 1st `  x )  =  (/)  \/  ( 1st `  x
)  =/=  (/) ) )
26 dcne 2266 . . . 4  |-  (DECID  ( 1st `  x )  =  (/)  <->  (
( 1st `  x
)  =  (/)  \/  ( 1st `  x )  =/=  (/) ) )
2725, 26sylibr 132 . . 3  |-  ( (
ph  /\  x  e.  ( A B ) )  -> DECID 
( 1st `  x
)  =  (/) )
288, 18, 27ifcldadc 3420 . 2  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  if ( ( 1st `  x )  =  (/) ,  ( F `
 ( 2nd `  x
) ) ,  ( G `  ( 2nd `  x ) ) )  e.  C )
29 updjudhf.h . 2  |-  H  =  ( x  e.  ( A B )  |->  if ( ( 1st `  x
)  =  (/) ,  ( F `  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )
3028, 29fmptd 5452 1  |-  ( ph  ->  H : ( A B ) --> C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438    =/= wne 2255   (/)c0 3286   ifcif 3393    |-> cmpt 3899   -->wf 5011   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   1oc1o 6174   ⊔ cdju 6730
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-fo 5021  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-dju 6731  df-inl 6739  df-inr 6740
This theorem is referenced by:  updjudhcoinlf  6771  updjudhcoinrg  6772  updjud  6773
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