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Theorem inrresf1 6947
Description: The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
inrresf1  |-  (inr  |`  B ) : B -1-1-> ( A B )

Proof of Theorem inrresf1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 djurf1or 6942 . 2  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
2 djurclr 6935 . 2  |-  ( x  e.  B  ->  (
(inr  |`  B ) `  x )  e.  ( A B ) )
31, 2inresflem 6945 1  |-  (inr  |`  B ) : B -1-1-> ( A B )
Colors of variables: wff set class
Syntax hints:    |` cres 4541   -1-1->wf1 5120   1oc1o 6306   ⊔ cdju 6922  inrcinr 6931
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inr 6933
This theorem is referenced by:  updjudhcoinrg  6966  updjud  6967  caserel  6972  djudom  6978  djufun  6989  djuinj  6991  djudomr  7076  exmidsbthrlem  13270
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