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Theorem inlresf1 9946
Description: The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
Assertion
Ref Expression
inlresf1 (inl ↾ š“):š“ā€“1-1→(š“ āŠ” šµ)

Proof of Theorem inlresf1
StepHypRef Expression
1 djulf1o 9943 . . 3 inl:V–1-1-onto→({āˆ…} Ɨ V)
2 f1of1 6843 . . 3 (inl:V–1-1-onto→({āˆ…} Ɨ V) → inl:V–1-1→({āˆ…} Ɨ V))
31, 2ax-mp 5 . 2 inl:V–1-1→({āˆ…} Ɨ V)
4 ssv 4006 . 2 š“ āŠ† V
5 inlresf 9945 . 2 (inl ↾ š“):š“āŸ¶(š“ āŠ” šµ)
6 f1resf1 6807 . 2 ((inl:V–1-1→({āˆ…} Ɨ V) ∧ š“ āŠ† V ∧ (inl ↾ š“):š“āŸ¶(š“ āŠ” šµ)) → (inl ↾ š“):š“ā€“1-1→(š“ āŠ” šµ))
73, 4, 5, 6mp3an 1457 1 (inl ↾ š“):š“ā€“1-1→(š“ āŠ” šµ)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3473   āŠ† wss 3949  āˆ…c0 4326  {csn 4632   Ɨ cxp 5680   ↾ cres 5684  āŸ¶wf 6549  ā€“1-1→wf1 6550  ā€“1-1-onto→wf1o 6552   āŠ” cdju 9929  inlcinl 9930
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
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 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1st 7999  df-2nd 8000  df-dju 9932  df-inl 9933
This theorem is referenced by: (None)
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