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Theorem inrresf1 7035
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 7030 . 2  |-  (inr  |`  B ) : B -1-1-onto-> ( { 1o }  X.  B )
2 djurclr 7023 . 2  |-  ( x  e.  B  ->  (
(inr  |`  B ) `  x )  e.  ( A B ) )
31, 2inresflem 7033 1  |-  (inr  |`  B ) : B -1-1-> ( A B )
Colors of variables: wff set class
Syntax hints:    |` cres 4611   -1-1->wf1 5193   1oc1o 6385   ⊔ cdju 7010  inrcinr 7019
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6116  df-2nd 6117  df-1o 6392  df-dju 7011  df-inr 7021
This theorem is referenced by:  updjudhcoinrg  7054  updjud  7055  caserel  7060  djudom  7066  djufun  7077  djuinj  7079  djudomr  7184  exmidsbthrlem  14014
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