Theorem casefun 6978

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 6951	. . . . . 6 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
3		f1of1 5374	. . . . . 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 5136	. . . . . 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 5171	. . . 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 6952	. . . . . 6 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
12		f1of1 5374	. . . . . 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 5136	. . . . . 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 5171	. . . 4 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
18	10, 16, 17	syl2anc 409	. . 3 |- ( ph -> Fun ( G o. `'inr ) )
19		dmcoss 4816	. . . . . . 7 |- dom ( F o. `'inl ) C_ dom `'inl
20		df-rn 4558	. . . . . . 7 |- ran inl = dom `'inl
21	19, 20	sseqtrri 3137	. . . . . 6 |- dom ( F o. `'inl ) C_ ran inl
22		dmcoss 4816	. . . . . . 7 |- dom ( G o. `'inr ) C_ dom `'inr
23		df-rn 4558	. . . . . . 7 |- ran inr = dom `'inr
24	22, 23	sseqtrri 3137	. . . . . 6 |- dom ( G o. `'inr ) C_ ran inr
25		ss2in 3309	. . . . . 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 5006	. . . . . . . . 9 |- ran (inl |` _V ) = ran inl
28	27	eqcomi 2144	. . . . . . . 8 |- ran inl = ran (inl |` _V )
29		rnresv 5006	. . . . . . . . 9 |- ran (inr |` _V ) = ran inr
30	29	eqcomi 2144	. . . . . . . 8 |- ran inr = ran (inr |` _V )
31	28, 30	ineq12i 3280	. . . . . . 7 |- ( ran inl i^i ran inr ) = ( ran (inl |` _V ) i^i ran (inr |` _V ) )
32		djuinr 6956	. . . . . . 7 |- ( ran (inl |` _V ) i^i ran (inr |` _V ) ) = (/)
33	31, 32	eqtri 2161	. . . . . 6 |- ( ran inl i^i ran inr ) = (/)
34	33	a1i 9	. . . . 5 |- ( ph -> ( ran inl i^i ran inr ) = (/) )
35	26, 34	sseqtrid 3152	. . . 4 |- ( ph -> ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ (/) )
36		ss0 3408	. . . 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 5175	. . 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 1216	. 2 |- ( ph -> Fun ( ( F o. `'inl ) u. ( G o. `'inr ) ) )
40		df-case 6977	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
41	40	funeqi 5152	. 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 1332   _Vcvv 2689   u. cun 3074   i^i cin 3075   C_ wss 3076   (/)c0 3368   {csn 3532   X. cxp 4545   `'ccnv 4546   dom cdm 4547   ran crn 4548   |` cres 4549   o. ccom 4551   Fun wfun 5125   -->wf 5127   -1-1->wf1 5128   -1-1-onto->wf1o 5130   1oc1o 6314  inlcinl 6938  inrcinr 6939  casecdjucase 6976

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-inl 6940  df-inr 6941  df-case 6977

This theorem is referenced by:  casef  6981