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Theorem inlresf1 7090
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 7085 . 2 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
2 djulclr 7078 . 2 (𝑥𝐴 → ((inl ↾ 𝐴)‘𝑥) ∈ (𝐴𝐵))
31, 2inresflem 7089 1 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  c0 3437  cres 4646  1-1wf1 5232  cdju 7066  inlcinl 7074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6165  df-2nd 6166  df-dju 7067  df-inl 7076
This theorem is referenced by:  updjudhcoinlf  7109  updjud  7111  caserel  7116  djudom  7122  difinfsn  7129  djufun  7133  djuinj  7135  djudoml  7248
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