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Theorem inrresf1 9030
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 9026 . . 3 inr:V–1-1-onto→({1𝑜} × V)
2 f1of1 6356 . . 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 3822 . 2 𝐵 ⊆ V
5 inrresf 9029 . 2 (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)
6 f1resf1 6325 . 2 ((inr:V–1-1→({1𝑜} × V) ∧ 𝐵 ⊆ V ∧ (inr ↾ 𝐵):𝐵⟶(𝐴𝐵)) → (inr ↾ 𝐵):𝐵1-1→(𝐴𝐵))
73, 4, 5, 6mp3an 1586 1 (inr ↾ 𝐵):𝐵1-1→(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3386  wss 3770  {csn 4369   × cxp 5311  cres 5315  wf 6098  1-1wf1 6099  1-1-ontowf1o 6101  1𝑜c1o 7793  cdju 9012  inrcinr 9014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-om 7301  df-1st 7402  df-2nd 7403  df-1o 7800  df-dju 9015  df-inr 9017
This theorem is referenced by: (None)
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