Theorem inlresf1 9811

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 9808	. . 3 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1of1 6763	. . 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 3960	. 2 ⊢ 𝐴 ⊆ V
5		inlresf 9810	. 2 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
6		f1resf1 6728	. 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 3903  ∅c0 4284  {csn 4577   × cxp 5617   ↾ cres 5621  ⟶wf 6478  –1-1→wf1 6479  –1-1-onto→wf1o 6481   ⊔ cdju 9794  inlcinl 9795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-1st 7924  df-2nd 7925  df-dju 9797  df-inl 9798

This theorem is referenced by: (None)