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Theorem inlresf 9943
Description: The left injection restricted to the left class of a disjoint union is a function from the left class into the disjoint union. (Contributed by AV, 27-Jun-2022.)
Assertion
Ref Expression
inlresf (inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡)

Proof of Theorem inlresf
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 djulf1o 9941 . . 3 inl:V–1-1-ontoβ†’({βˆ…} Γ— V)
2 f1ofun 6844 . . 3 (inl:V–1-1-ontoβ†’({βˆ…} Γ— V) β†’ Fun inl)
3 ffvresb 7138 . . 3 (Fun inl β†’ ((inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡) ↔ βˆ€π‘₯ ∈ 𝐴 (π‘₯ ∈ dom inl ∧ (inlβ€˜π‘₯) ∈ (𝐴 βŠ” 𝐡))))
41, 2, 3mp2b 10 . 2 ((inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡) ↔ βˆ€π‘₯ ∈ 𝐴 (π‘₯ ∈ dom inl ∧ (inlβ€˜π‘₯) ∈ (𝐴 βŠ” 𝐡)))
5 elex 3490 . . . 4 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ V)
6 opex 5468 . . . . 5 βŸ¨βˆ…, π‘₯⟩ ∈ V
7 df-inl 9931 . . . . 5 inl = (π‘₯ ∈ V ↦ βŸ¨βˆ…, π‘₯⟩)
86, 7dmmpti 6702 . . . 4 dom inl = V
95, 8eleqtrrdi 2839 . . 3 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ dom inl)
10 djulcl 9939 . . 3 (π‘₯ ∈ 𝐴 β†’ (inlβ€˜π‘₯) ∈ (𝐴 βŠ” 𝐡))
119, 10jca 510 . 2 (π‘₯ ∈ 𝐴 β†’ (π‘₯ ∈ dom inl ∧ (inlβ€˜π‘₯) ∈ (𝐴 βŠ” 𝐡)))
124, 11mprgbir 3064 1 (inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   ∈ wcel 2098  βˆ€wral 3057  Vcvv 3471  βˆ…c0 4324  {csn 4630  βŸ¨cop 4636   Γ— cxp 5678  dom cdm 5680   β†Ύ cres 5682  Fun wfun 6545  βŸΆwf 6547  β€“1-1-ontoβ†’wf1o 6550  β€˜cfv 6551   βŠ” cdju 9927  inlcinl 9928
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-1st 7997  df-2nd 7998  df-dju 9930  df-inl 9931
This theorem is referenced by:  inlresf1  9944  updjudhcoinlf  9961  updjud  9963
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