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Theorem inlresf1 9190
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 9187 . . 3 inl:V–1-1-onto→({∅} × V)
2 f1of1 6482 . . 3 (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
31, 2ax-mp 5 . 2 inl:V–1-1→({∅} × V)
4 ssv 3912 . 2 𝐴 ⊆ V
5 inlresf 9189 . 2 (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)
6 f1resf1 6451 . 2 ((inl:V–1-1→({∅} × V) ∧ 𝐴 ⊆ V ∧ (inl ↾ 𝐴):𝐴⟶(𝐴𝐵)) → (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵))
73, 4, 5, 6mp3an 1453 1 (inl ↾ 𝐴):𝐴1-1→(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3437  wss 3859  c0 4211  {csn 4472   × cxp 5441  cres 5445  wf 6221  1-1wf1 6222  1-1-ontowf1o 6224  cdju 9173  inlcinl 9174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-1st 7545  df-2nd 7546  df-dju 9176  df-inl 9177
This theorem is referenced by: (None)
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