Theorem casef1 7381

Description: The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
casef1.f	|- ( ph -> F : A -1-1-> X )
casef1.g	|- ( ph -> G : B -1-1-> X )
casef1.disj	|- ( ph -> ( ran F i^i ran G ) = (/) )

Assertion

Ref	Expression
casef1	|- ( ph -> case ( F , G ) : ( A ⊔ B ) -1-1-> X )

Proof of Theorem casef1

Step	Hyp	Ref	Expression
1		casef1.f	. . . 4 |- ( ph -> F : A -1-1-> X )
2		f1f 5573	. . . 4 |- ( F : A -1-1-> X -> F : A --> X )
3	1, 2	syl 14	. . 3 |- ( ph -> F : A --> X )
4		casef1.g	. . . 4 |- ( ph -> G : B -1-1-> X )
5		f1f 5573	. . . 4 |- ( G : B -1-1-> X -> G : B --> X )
6	4, 5	syl 14	. . 3 |- ( ph -> G : B --> X )
7	3, 6	casef 7379	. 2 |- ( ph -> case ( F , G ) : ( A ⊔ B ) --> X )
8		df-f1 5357	. . . . 5 |- ( F : A -1-1-> X <-> ( F : A --> X /\ Fun `' F ) )
9	8	simprbi 275	. . . 4 |- ( F : A -1-1-> X -> Fun `' F )
10	1, 9	syl 14	. . 3 |- ( ph -> Fun `' F )
11		df-f1 5357	. . . . 5 |- ( G : B -1-1-> X <-> ( G : B --> X /\ Fun `' G ) )
12	11	simprbi 275	. . . 4 |- ( G : B -1-1-> X -> Fun `' G )
13	4, 12	syl 14	. . 3 |- ( ph -> Fun `' G )
14		casef1.disj	. . 3 |- ( ph -> ( ran F i^i ran G ) = (/) )
15	10, 13, 14	caseinj 7380	. 2 |- ( ph -> Fun `'case ( F , G ) )
16		df-f1 5357	. 2 |- (case ( F , G ) : ( A ⊔ B ) -1-1-> X <-> (case ( F , G ) : ( A ⊔ B ) --> X /\ Fun `'case ( F , G ) ) )
17	7, 15, 16	sylanbrc 417	1 |- ( ph -> case ( F , G ) : ( A ⊔ B ) -1-1-> X )

Syntax hints:   -> wi 4   = wceq 1398   i^i cin 3210   (/)c0 3508   `'ccnv 4748   ran crn 4750   Fun wfun 5346   -->wf 5348   -1-1->wf1 5349   ⊔ cdju 7328  casecdjucase 7374

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-fal 1404  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-ne 2413  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-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  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-1o 6647  df-dju 7329  df-inl 7338  df-inr 7339  df-case 7375

This theorem is referenced by:  djudom  7384  exmidsbthrlem  16802