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Theorem inlresf1 6946
Description: The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
inlresf1 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)

Proof of Theorem inlresf1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 djulf1or 6941 . 2 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
2 djulclr 6934 . 2 (𝑥𝐴 → ((inl ↾ 𝐴)‘𝑥) ∈ (𝐴𝐵))
31, 2inresflem 6945 1 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  c0 3363  cres 4541  1-1wf1 5120  cdju 6922  inlcinl 6930
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-id 4215  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-dju 6923  df-inl 6932
This theorem is referenced by:  updjudhcoinlf  6965  updjud  6967  caserel  6972  djudom  6978  difinfsn  6985  djufun  6989  djuinj  6991  djudoml  7080
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