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Theorem inlresf1 9377
 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
StepHypRef Expression
1 djulf1o 9374 . . 3 inl:V–1-1-onto→({∅} × V)
2 f1of1 6601 . . 3 (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
31, 2ax-mp 5 . 2 inl:V–1-1→({∅} × V)
4 ssv 3916 . 2 𝐴 ⊆ V
5 inlresf 9376 . 2 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
6 f1resf1 6569 . 2 ((inl:V–1-1→({∅} × V) ∧ 𝐴 ⊆ V ∧ (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)) → (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵))
73, 4, 5, 6mp3an 1458 1 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  {csn 4522   × cxp 5522   ↾ cres 5526  ⟶wf 6331  –1-1→wf1 6332  –1-1-onto→wf1o 6334   ⊔ cdju 9360  inlcinl 9361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-1st 7693  df-2nd 7694  df-dju 9363  df-inl 9364 This theorem is referenced by: (None)
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