Theorem inlresf1 9673

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 ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)

Proof of Theorem inlresf1

Step	Hyp	Ref	Expression
1		djulf1o 9670	. . 3 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1of1 6715	. . 3 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
3	1, 2	ax-mp 5	. 2 ⊢ inl:V–1-1→({∅} × V)
4		ssv 3945	. 2 ⊢ 𝐴 ⊆ V
5		inlresf 9672	. 2 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
6		f1resf1 6679	. 2 ⊢ ((inl:V–1-1→({∅} × V) ∧ 𝐴 ⊆ V ∧ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)) → (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵))
7	3, 4, 5, 6	mp3an 1460	1 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)

Syntax hints:  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  {csn 4561   × cxp 5587   ↾ cres 5591  ⟶wf 6429  –1-1→wf1 6430  –1-1-onto→wf1o 6432   ⊔ cdju 9656  inlcinl 9657

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1st 7831  df-2nd 7832  df-dju 9659  df-inl 9660

This theorem is referenced by: (None)