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Theorem inlresf1 6751
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 6746 . 2 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
2 djulclr 6739 . 2 (𝑥𝐴 → ((inl ↾ 𝐴)‘𝑥) ∈ (𝐴𝐵))
31, 2inresflem 6750 1 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  c0 3286  cres 4440  1-1wf1 5012  cdju 6728  inlcinl 6735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-dju 6729  df-inl 6737
This theorem is referenced by:  djuun  6758  updjudhcoinlf  6769  updjud  6771  caserel  6776  djufun  6782  djuinj  6784  djudom  6785
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