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Theorem inrresf1 7039
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
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 djurf1or 7034 . 2 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
2 djurclr 7027 . 2 (𝑥𝐵 → ((inr ↾ 𝐵)‘𝑥) ∈ (𝐴𝐵))
31, 2inresflem 7037 1 (inr ↾ 𝐵):𝐵1-1→(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  cres 4613  1-1wf1 5195  1oc1o 6388  cdju 7014  inrcinr 7023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inr 7025
This theorem is referenced by:  updjudhcoinrg  7058  updjud  7059  caserel  7064  djudom  7070  djufun  7081  djuinj  7083  djudomr  7197  exmidsbthrlem  14054
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