Theorem inlresf1 9808

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 9805	. . 3 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1of1 6762	. . 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 3954	. 2 ⊢ 𝐴 ⊆ V
5		inlresf 9807	. 2 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
6		f1resf1 6727	. 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 1463	1 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)

Syntax hints:  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  {csn 4573   × cxp 5612   ↾ cres 5616  ⟶wf 6477  –1-1→wf1 6478  –1-1-onto→wf1o 6480   ⊔ cdju 9791  inlcinl 9792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-1st 7921  df-2nd 7922  df-dju 9794  df-inl 9795

This theorem is referenced by: (None)