Theorem inlresf1 7189

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

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djulf1or 7184	. 2 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
2		djulclr 7177	. 2 ⊢ (𝑥 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑥) ∈ (𝐴 ⊔ 𝐵))
3	1, 2	inresflem 7188	1 ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)

Syntax hints:  ∅c0 3468   ↾ cres 4695  –1-1→wf1 5287   ⊔ cdju 7165  inlcinl 7173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-dju 7166  df-inl 7175

This theorem is referenced by:  updjudhcoinlf  7208  updjud  7210  caserel  7215  djudom  7221  difinfsn  7228  djufun  7232  djuinj  7234  djudoml  7362