Theorem inlresf1 7352

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 7347	. 2 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		djulclr 7340	. 2 |- ( x e. A -> ( (inl |` A ) ` x ) e. ( A ⊔ B ) )
3	1, 2	inresflem 7351	1 |- (inl |` A ) : A -1-1-> ( A ⊔ B )

Syntax hints:   (/)c0 3508   |` cres 4751   -1-1->wf1 5349   ⊔ cdju 7328  inlcinl 7336

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1st 6334  df-2nd 6335  df-dju 7329  df-inl 7338

This theorem is referenced by:  updjudhcoinlf  7371  updjud  7373  caserel  7378  djudom  7384  difinfsn  7391  djufun  7395  djuinj  7397  djudoml  7526