Theorem casefun 7252

Description: The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
casefun.f	|- ( ph -> Fun F )
casefun.g	|- ( ph -> Fun G )

Assertion

Ref	Expression
casefun	|- ( ph -> Fun case ( F , G ) )

Proof of Theorem casefun

Step	Hyp	Ref	Expression
1		casefun.f	. . . 4 |- ( ph -> Fun F )
2		djulf1o 7225	. . . . . 6 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
3		f1of1 5571	. . . . . 6 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> inl : _V -1-1-> ( { (/) } X. _V ) )
4	2, 3	ax-mp 5	. . . . 5 |- inl : _V -1-1-> ( { (/) } X. _V )
5		df-f1 5323	. . . . . 6 |- (inl : _V -1-1-> ( { (/) } X. _V ) <-> (inl : _V --> ( { (/) } X. _V ) /\ Fun `'inl ) )
6	5	simprbi 275	. . . . 5 |- (inl : _V -1-1-> ( { (/) } X. _V ) -> Fun `'inl )
7	4, 6	mp1i 10	. . . 4 |- ( ph -> Fun `'inl )
8		funco 5358	. . . 4 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
9	1, 7, 8	syl2anc 411	. . 3 |- ( ph -> Fun ( F o. `'inl ) )
10		casefun.g	. . . 4 |- ( ph -> Fun G )
11		djurf1o 7226	. . . . . 6 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
12		f1of1 5571	. . . . . 6 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> inr : _V -1-1-> ( { 1o } X. _V ) )
13	11, 12	ax-mp 5	. . . . 5 |- inr : _V -1-1-> ( { 1o } X. _V )
14		df-f1 5323	. . . . . 6 |- (inr : _V -1-1-> ( { 1o } X. _V ) <-> (inr : _V --> ( { 1o } X. _V ) /\ Fun `'inr ) )
15	14	simprbi 275	. . . . 5 |- (inr : _V -1-1-> ( { 1o } X. _V ) -> Fun `'inr )
16	13, 15	mp1i 10	. . . 4 |- ( ph -> Fun `'inr )
17		funco 5358	. . . 4 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
18	10, 16, 17	syl2anc 411	. . 3 |- ( ph -> Fun ( G o. `'inr ) )
19		dmcoss 4994	. . . . . . 7 |- dom ( F o. `'inl ) C_ dom `'inl
20		df-rn 4730	. . . . . . 7 |- ran inl = dom `'inl
21	19, 20	sseqtrri 3259	. . . . . 6 |- dom ( F o. `'inl ) C_ ran inl
22		dmcoss 4994	. . . . . . 7 |- dom ( G o. `'inr ) C_ dom `'inr
23		df-rn 4730	. . . . . . 7 |- ran inr = dom `'inr
24	22, 23	sseqtrri 3259	. . . . . 6 |- dom ( G o. `'inr ) C_ ran inr
25		ss2in 3432	. . . . . 6 |- ( ( dom ( F o. `'inl ) C_ ran inl /\ dom ( G o. `'inr ) C_ ran inr ) -> ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ ( ran inl i^i ran inr ) )
26	21, 24, 25	mp2an 426	. . . . 5 |- ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ ( ran inl i^i ran inr )
27		rnresv 5188	. . . . . . . . 9 |- ran (inl |` _V ) = ran inl
28	27	eqcomi 2233	. . . . . . . 8 |- ran inl = ran (inl |` _V )
29		rnresv 5188	. . . . . . . . 9 |- ran (inr |` _V ) = ran inr
30	29	eqcomi 2233	. . . . . . . 8 |- ran inr = ran (inr |` _V )
31	28, 30	ineq12i 3403	. . . . . . 7 |- ( ran inl i^i ran inr ) = ( ran (inl |` _V ) i^i ran (inr |` _V ) )
32		djuinr 7230	. . . . . . 7 |- ( ran (inl |` _V ) i^i ran (inr |` _V ) ) = (/)
33	31, 32	eqtri 2250	. . . . . 6 |- ( ran inl i^i ran inr ) = (/)
34	33	a1i 9	. . . . 5 |- ( ph -> ( ran inl i^i ran inr ) = (/) )
35	26, 34	sseqtrid 3274	. . . 4 |- ( ph -> ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ (/) )
36		ss0 3532	. . . 4 |- ( ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ (/) -> ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) = (/) )
37	35, 36	syl 14	. . 3 |- ( ph -> ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) = (/) )
38		funun 5362	. . 3 |- ( ( ( Fun ( F o. `'inl ) /\ Fun ( G o. `'inr ) ) /\ ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) = (/) ) -> Fun ( ( F o. `'inl ) u. ( G o. `'inr ) ) )
39	9, 18, 37, 38	syl21anc 1270	. 2 |- ( ph -> Fun ( ( F o. `'inl ) u. ( G o. `'inr ) ) )
40		df-case 7251	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
41	40	funeqi 5339	. 2 |- ( Fun case ( F , G ) <-> Fun ( ( F o. `'inl ) u. ( G o. `'inr ) ) )
42	39, 41	sylibr 134	1 |- ( ph -> Fun case ( F , G ) )

Syntax hints:   -> wi 4   = wceq 1395   _Vcvv 2799   u. cun 3195   i^i cin 3196   C_ wss 3197   (/)c0 3491   {csn 3666   X. cxp 4717   `'ccnv 4718   dom cdm 4719   ran crn 4720   |` cres 4721   o. ccom 4723   Fun wfun 5312   -->wf 5314   -1-1->wf1 5315   -1-1-onto->wf1o 5317   1oc1o 6555  inlcinl 7212  inrcinr 7213  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-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-inl 7214  df-inr 7215  df-case 7251

This theorem is referenced by:  casef  7255