Theorem inlresf1 9831

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 9828	. . 3 ⊢ inl:V–1-1-onto→({∅} × V)
2		f1of1 6767	. . 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 3939	. 2 ⊢ 𝐴 ⊆ V
5		inlresf 9830	. 2 ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵)
6		f1resf1 6732	. 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 1469	1 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)

Syntax hints:  Vcvv 3431   ⊆ wss 3883  ∅c0 4262  {csn 4556   × cxp 5617   ↾ cres 5621  ⟶wf 6482  –1-1→wf1 6483  –1-1-onto→wf1o 6485   ⊔ cdju 9814  inlcinl 9815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-1st 7932  df-2nd 7933  df-dju 9817  df-inl 9818

This theorem is referenced by: (None)