Theorem inlresf1 7165

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 7160	. 2 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		djulclr 7153	. 2 |- ( x e. A -> ( (inl |` A ) ` x ) e. ( A ⊔ B ) )
3	1, 2	inresflem 7164	1 |- (inl |` A ) : A -1-1-> ( A ⊔ B )

Syntax hints:   (/)c0 3460   |` cres 4678   -1-1->wf1 5269   ⊔ cdju 7141  inlcinl 7149

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-1st 6228  df-2nd 6229  df-dju 7142  df-inl 7151

This theorem is referenced by:  updjudhcoinlf  7184  updjud  7186  caserel  7191  djudom  7197  difinfsn  7204  djufun  7208  djuinj  7210  djudoml  7333