Theorem inlresf1 7259

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 7254	. 2 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		djulclr 7247	. 2 |- ( x e. A -> ( (inl |` A ) ` x ) e. ( A ⊔ B ) )
3	1, 2	inresflem 7258	1 |- (inl |` A ) : A -1-1-> ( A ⊔ B )

Syntax hints:   (/)c0 3494   |` cres 4727   -1-1->wf1 5323   ⊔ cdju 7235  inlcinl 7243

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-dju 7236  df-inl 7245

This theorem is referenced by:  updjudhcoinlf  7278  updjud  7280  caserel  7285  djudom  7291  difinfsn  7298  djufun  7302  djuinj  7304  djudoml  7433