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Theorem updjudhf 6930
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 6923 . . . . . 6  |-  ( ( x  e.  ( A B )  /\  ( 1st `  x )  =  (/) )  ->  ( 2nd `  x )  e.  A
)
21ex 114 . . . . 5  |-  ( x  e.  ( A B )  ->  ( ( 1st `  x )  =  (/)  ->  ( 2nd `  x
)  e.  A ) )
3 updjud.f . . . . . 6  |-  ( ph  ->  F : A --> C )
4 ffvelrn 5519 . . . . . . 7  |-  ( ( F : A --> C  /\  ( 2nd `  x )  e.  A )  -> 
( F `  ( 2nd `  x ) )  e.  C )
54ex 114 . . . . . 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 405 . . . 4  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  ( ( 1st `  x )  =  (/)  ->  ( F `  ( 2nd `  x ) )  e.  C ) )
87imp 123 . . 3  |-  ( ( ( ph  /\  x  e.  ( A B )
)  /\  ( 1st `  x )  =  (/) )  ->  ( F `  ( 2nd `  x ) )  e.  C )
9 df-ne 2284 . . . . 5  |-  ( ( 1st `  x )  =/=  (/)  <->  -.  ( 1st `  x )  =  (/) )
10 eldju2ndr 6924 . . . . . . 7  |-  ( ( x  e.  ( A B )  /\  ( 1st `  x )  =/=  (/) )  ->  ( 2nd `  x )  e.  B
)
1110ex 114 . . . . . 6  |-  ( x  e.  ( A B )  ->  ( ( 1st `  x )  =/=  (/)  ->  ( 2nd `  x )  e.  B ) )
12 updjud.g . . . . . . 7  |-  ( ph  ->  G : B --> C )
13 ffvelrn 5519 . . . . . . . 8  |-  ( ( G : B --> C  /\  ( 2nd `  x )  e.  B )  -> 
( G `  ( 2nd `  x ) )  e.  C )
1413ex 114 . . . . . . 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 405 . . . . 5  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  ( ( 1st `  x )  =/=  (/)  ->  ( G `  ( 2nd `  x ) )  e.  C ) )
179, 16syl5bir 152 . . . 4  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  ( -.  ( 1st `  x )  =  (/)  ->  ( G `  ( 2nd `  x ) )  e.  C ) )
1817imp 123 . . 3  |-  ( ( ( ph  /\  x  e.  ( A B )
)  /\  -.  ( 1st `  x )  =  (/) )  ->  ( G `
 ( 2nd `  x
) )  e.  C
)
19 eldju1st 6922 . . . . . 6  |-  ( x  e.  ( A B )  ->  ( ( 1st `  x )  =  (/)  \/  ( 1st `  x
)  =  1o ) )
20 1n0 6295 . . . . . . . 8  |-  1o  =/=  (/)
21 neeq1 2296 . . . . . . . 8  |-  ( ( 1st `  x )  =  1o  ->  (
( 1st `  x
)  =/=  (/)  <->  1o  =/=  (/) ) )
2220, 21mpbiri 167 . . . . . . 7  |-  ( ( 1st `  x )  =  1o  ->  ( 1st `  x )  =/=  (/) )
2322orim2i 733 . . . . . 6  |-  ( ( ( 1st `  x
)  =  (/)  \/  ( 1st `  x )  =  1o )  ->  (
( 1st `  x
)  =  (/)  \/  ( 1st `  x )  =/=  (/) ) )
2419, 23syl 14 . . . . 5  |-  ( x  e.  ( A B )  ->  ( ( 1st `  x )  =  (/)  \/  ( 1st `  x
)  =/=  (/) ) )
2524adantl 273 . . . 4  |-  ( (
ph  /\  x  e.  ( A B ) )  ->  ( ( 1st `  x )  =  (/)  \/  ( 1st `  x
)  =/=  (/) ) )
26 dcne 2294 . . . 4  |-  (DECID  ( 1st `  x )  =  (/)  <->  (
( 1st `  x
)  =  (/)  \/  ( 1st `  x )  =/=  (/) ) )
2725, 26sylibr 133 . . 3  |-  ( (
ph  /\  x  e.  ( A B ) )  -> DECID 
( 1st `  x
)  =  (/) )
288, 18, 27ifcldadc 3469 . 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 5540 1  |-  ( ph  ->  H : ( A B ) --> C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463    =/= wne 2283   (/)c0 3331   ifcif 3442    |-> cmpt 3957   -->wf 5087   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   1oc1o 6272   ⊔ cdju 6888
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by:  updjudhcoinlf  6931  updjudhcoinrg  6932  updjud  6933
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