Theorem casefun 7050

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 7023	. . . . . 6 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
3		f1of1 5431	. . . . . 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 5193	. . . . . 6 |- (inl : _V -1-1-> ( { (/) } X. _V ) <-> (inl : _V --> ( { (/) } X. _V ) /\ Fun `'inl ) )
6	5	simprbi 273	. . . . 5 |- (inl : _V -1-1-> ( { (/) } X. _V ) -> Fun `'inl )
7	4, 6	mp1i 10	. . . 4 |- ( ph -> Fun `'inl )
8		funco 5228	. . . 4 |- ( ( Fun F /\ Fun `'inl ) -> Fun ( F o. `'inl ) )
9	1, 7, 8	syl2anc 409	. . 3 |- ( ph -> Fun ( F o. `'inl ) )
10		casefun.g	. . . 4 |- ( ph -> Fun G )
11		djurf1o 7024	. . . . . 6 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
12		f1of1 5431	. . . . . 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 5193	. . . . . 6 |- (inr : _V -1-1-> ( { 1o } X. _V ) <-> (inr : _V --> ( { 1o } X. _V ) /\ Fun `'inr ) )
15	14	simprbi 273	. . . . 5 |- (inr : _V -1-1-> ( { 1o } X. _V ) -> Fun `'inr )
16	13, 15	mp1i 10	. . . 4 |- ( ph -> Fun `'inr )
17		funco 5228	. . . 4 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
18	10, 16, 17	syl2anc 409	. . 3 |- ( ph -> Fun ( G o. `'inr ) )
19		dmcoss 4873	. . . . . . 7 |- dom ( F o. `'inl ) C_ dom `'inl
20		df-rn 4615	. . . . . . 7 |- ran inl = dom `'inl
21	19, 20	sseqtrri 3177	. . . . . 6 |- dom ( F o. `'inl ) C_ ran inl
22		dmcoss 4873	. . . . . . 7 |- dom ( G o. `'inr ) C_ dom `'inr
23		df-rn 4615	. . . . . . 7 |- ran inr = dom `'inr
24	22, 23	sseqtrri 3177	. . . . . 6 |- dom ( G o. `'inr ) C_ ran inr
25		ss2in 3350	. . . . . 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 423	. . . . 5 |- ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ ( ran inl i^i ran inr )
27		rnresv 5063	. . . . . . . . 9 |- ran (inl |` _V ) = ran inl
28	27	eqcomi 2169	. . . . . . . 8 |- ran inl = ran (inl |` _V )
29		rnresv 5063	. . . . . . . . 9 |- ran (inr |` _V ) = ran inr
30	29	eqcomi 2169	. . . . . . . 8 |- ran inr = ran (inr |` _V )
31	28, 30	ineq12i 3321	. . . . . . 7 |- ( ran inl i^i ran inr ) = ( ran (inl |` _V ) i^i ran (inr |` _V ) )
32		djuinr 7028	. . . . . . 7 |- ( ran (inl |` _V ) i^i ran (inr |` _V ) ) = (/)
33	31, 32	eqtri 2186	. . . . . 6 |- ( ran inl i^i ran inr ) = (/)
34	33	a1i 9	. . . . 5 |- ( ph -> ( ran inl i^i ran inr ) = (/) )
35	26, 34	sseqtrid 3192	. . . 4 |- ( ph -> ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ (/) )
36		ss0 3449	. . . 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 5232	. . 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 1227	. 2 |- ( ph -> Fun ( ( F o. `'inl ) u. ( G o. `'inr ) ) )
40		df-case 7049	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
41	40	funeqi 5209	. 2 |- ( Fun case ( F , G ) <-> Fun ( ( F o. `'inl ) u. ( G o. `'inr ) ) )
42	39, 41	sylibr 133	1 |- ( ph -> Fun case ( F , G ) )

Syntax hints:   -> wi 4   = wceq 1343   _Vcvv 2726   u. cun 3114   i^i cin 3115   C_ wss 3116   (/)c0 3409   {csn 3576   X. cxp 4602   `'ccnv 4603   dom cdm 4604   ran crn 4605   |` cres 4606   o. ccom 4608   Fun wfun 5182   -->wf 5184   -1-1->wf1 5185   -1-1-onto->wf1o 5187   1oc1o 6377  inlcinl 7010  inrcinr 7011  casecdjucase 7048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-inl 7012  df-inr 7013  df-case 7049

This theorem is referenced by:  casef  7053