Theorem casef1 7257

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 5531	. . . 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 5531	. . . 4 |- ( G : B -1-1-> X -> G : B --> X )
6	4, 5	syl 14	. . 3 |- ( ph -> G : B --> X )
7	3, 6	casef 7255	. 2 |- ( ph -> case ( F , G ) : ( A ⊔ B ) --> X )
8		df-f1 5323	. . . . 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 5323	. . . . 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 7256	. 2 |- ( ph -> Fun `'case ( F , G ) )
16		df-f1 5323	. 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 1395   i^i cin 3196   (/)c0 3491   `'ccnv 4718   ran crn 4720   Fun wfun 5312   -->wf 5314   -1-1->wf1 5315   ⊔ cdju 7204  casecdjucase 7250

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6286  df-2nd 6287  df-1o 6562  df-dju 7205  df-inl 7214  df-inr 7215  df-case 7251

This theorem is referenced by:  djudom  7260  exmidsbthrlem  16390