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Theorem inrresf1 9338
Description: The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
inrresf1 (inr ↾ 𝐵):𝐵1-1→(𝐴𝐵)

Proof of Theorem inrresf1
StepHypRef Expression
1 djurf1o 9334 . . 3 inr:V–1-1-onto→({1o} × V)
2 f1of1 6610 . . 3 (inr:V–1-1-onto→({1o} × V) → inr:V–1-1→({1o} × V))
31, 2ax-mp 5 . 2 inr:V–1-1→({1o} × V)
4 ssv 3994 . 2 𝐵 ⊆ V
5 inrresf 9337 . 2 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
6 f1resf1 6579 . 2 ((inr:V–1-1→({1o} × V) ∧ 𝐵 ⊆ V ∧ (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)) → (inr ↾ 𝐵):𝐵1-1→(𝐴𝐵))
73, 4, 5, 6mp3an 1454 1 (inr ↾ 𝐵):𝐵1-1→(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3499  wss 3939  {csn 4563   × cxp 5551  cres 5555  wf 6347  1-1wf1 6348  1-1-ontowf1o 6350  1oc1o 8089  cdju 9319  inrcinr 9321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-1st 7683  df-2nd 7684  df-1o 8096  df-dju 9322  df-inr 9324
This theorem is referenced by: (None)
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