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Theorem casef1 6943
Description: The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
casef1.f  |-  ( ph  ->  F : A -1-1-> X
)
casef1.g  |-  ( ph  ->  G : B -1-1-> X
)
casef1.disj  |-  ( ph  ->  ( ran  F  i^i  ran 
G )  =  (/) )
Assertion
Ref Expression
casef1  |-  ( ph  -> case ( F ,  G
) : ( A B ) -1-1-> X )

Proof of Theorem casef1
StepHypRef Expression
1 casef1.f . . . 4  |-  ( ph  ->  F : A -1-1-> X
)
2 f1f 5298 . . . 4  |-  ( F : A -1-1-> X  ->  F : A --> X )
31, 2syl 14 . . 3  |-  ( ph  ->  F : A --> X )
4 casef1.g . . . 4  |-  ( ph  ->  G : B -1-1-> X
)
5 f1f 5298 . . . 4  |-  ( G : B -1-1-> X  ->  G : B --> X )
64, 5syl 14 . . 3  |-  ( ph  ->  G : B --> X )
73, 6casef 6941 . 2  |-  ( ph  -> case ( F ,  G
) : ( A B ) --> X )
8 df-f1 5098 . . . . 5  |-  ( F : A -1-1-> X  <->  ( F : A --> X  /\  Fun  `' F ) )
98simprbi 273 . . . 4  |-  ( F : A -1-1-> X  ->  Fun  `' F )
101, 9syl 14 . . 3  |-  ( ph  ->  Fun  `' F )
11 df-f1 5098 . . . . 5  |-  ( G : B -1-1-> X  <->  ( G : B --> X  /\  Fun  `' G ) )
1211simprbi 273 . . . 4  |-  ( G : B -1-1-> X  ->  Fun  `' G )
134, 12syl 14 . . 3  |-  ( ph  ->  Fun  `' G )
14 casef1.disj . . 3  |-  ( ph  ->  ( ran  F  i^i  ran 
G )  =  (/) )
1510, 13, 14caseinj 6942 . 2  |-  ( ph  ->  Fun  `'case ( F ,  G
) )
16 df-f1 5098 . 2  |-  (case ( F ,  G ) : ( A B )
-1-1-> X  <->  (case ( F ,  G ) : ( A B ) --> X  /\  Fun  `'case ( F ,  G
) ) )
177, 15, 16sylanbrc 413 1  |-  ( ph  -> case ( F ,  G
) : ( A B ) -1-1-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    i^i cin 3040   (/)c0 3333   `'ccnv 4508   ran crn 4510   Fun wfun 5087   -->wf 5089   -1-1->wf1 5090   ⊔ cdju 6890  casecdjucase 6936
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901  df-case 6937
This theorem is referenced by:  djudom  6946  exmidsbthrlem  13113
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