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Theorem inlresf 9908
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 9906 . . 3 inl:V–1-1-ontoβ†’({βˆ…} Γ— V)
2 f1ofun 6828 . . 3 (inl:V–1-1-ontoβ†’({βˆ…} Γ— V) β†’ Fun inl)
3 ffvresb 7119 . . 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 3487 . . . 4 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ V)
6 opex 5457 . . . . 5 βŸ¨βˆ…, π‘₯⟩ ∈ V
7 df-inl 9896 . . . . 5 inl = (π‘₯ ∈ V ↦ βŸ¨βˆ…, π‘₯⟩)
86, 7dmmpti 6687 . . . 4 dom inl = V
95, 8eleqtrrdi 2838 . . 3 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ dom inl)
10 djulcl 9904 . . 3 (π‘₯ ∈ 𝐴 β†’ (inlβ€˜π‘₯) ∈ (𝐴 βŠ” 𝐡))
119, 10jca 511 . 2 (π‘₯ ∈ 𝐴 β†’ (π‘₯ ∈ dom inl ∧ (inlβ€˜π‘₯) ∈ (𝐴 βŠ” 𝐡)))
124, 11mprgbir 3062 1 (inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468  βˆ…c0 4317  {csn 4623  βŸ¨cop 4629   Γ— cxp 5667  dom cdm 5669   β†Ύ cres 5671  Fun wfun 6530  βŸΆwf 6532  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536   βŠ” cdju 9892  inlcinl 9893
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1st 7971  df-2nd 7972  df-dju 9895  df-inl 9896
This theorem is referenced by:  inlresf1  9909  updjudhcoinlf  9926  updjud  9928
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