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Theorem inlresf 9858
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 9856 . . 3 inl:V–1-1-ontoβ†’({βˆ…} Γ— V)
2 f1ofun 6790 . . 3 (inl:V–1-1-ontoβ†’({βˆ…} Γ— V) β†’ Fun inl)
3 ffvresb 7076 . . 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 3465 . . . 4 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ V)
6 opex 5425 . . . . 5 βŸ¨βˆ…, π‘₯⟩ ∈ V
7 df-inl 9846 . . . . 5 inl = (π‘₯ ∈ V ↦ βŸ¨βˆ…, π‘₯⟩)
86, 7dmmpti 6649 . . . 4 dom inl = V
95, 8eleqtrrdi 2845 . . 3 (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ dom inl)
10 djulcl 9854 . . 3 (π‘₯ ∈ 𝐴 β†’ (inlβ€˜π‘₯) ∈ (𝐴 βŠ” 𝐡))
119, 10jca 513 . 2 (π‘₯ ∈ 𝐴 β†’ (π‘₯ ∈ dom inl ∧ (inlβ€˜π‘₯) ∈ (𝐴 βŠ” 𝐡)))
124, 11mprgbir 3068 1 (inl β†Ύ 𝐴):𝐴⟢(𝐴 βŠ” 𝐡)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447  βˆ…c0 4286  {csn 4590  βŸ¨cop 4596   Γ— cxp 5635  dom cdm 5637   β†Ύ cres 5639  Fun wfun 6494  βŸΆwf 6496  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500   βŠ” cdju 9842  inlcinl 9843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1st 7925  df-2nd 7926  df-dju 9845  df-inl 9846
This theorem is referenced by:  inlresf1  9859  updjudhcoinlf  9876  updjud  9878
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