Theorem casefun 7289

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 7262	. . . . . 6 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
3		f1of1 5585	. . . . . 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 5333	. . . . . 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 5368	. . . 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 7263	. . . . . 6 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
12		f1of1 5585	. . . . . 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 5333	. . . . . 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 5368	. . . 4 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
18	10, 16, 17	syl2anc 411	. . 3 |- ( ph -> Fun ( G o. `'inr ) )
19		dmcoss 5004	. . . . . . 7 |- dom ( F o. `'inl ) C_ dom `'inl
20		df-rn 4738	. . . . . . 7 |- ran inl = dom `'inl
21	19, 20	sseqtrri 3261	. . . . . 6 |- dom ( F o. `'inl ) C_ ran inl
22		dmcoss 5004	. . . . . . 7 |- dom ( G o. `'inr ) C_ dom `'inr
23		df-rn 4738	. . . . . . 7 |- ran inr = dom `'inr
24	22, 23	sseqtrri 3261	. . . . . 6 |- dom ( G o. `'inr ) C_ ran inr
25		ss2in 3434	. . . . . 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 5198	. . . . . . . . 9 |- ran (inl |` _V ) = ran inl
28	27	eqcomi 2234	. . . . . . . 8 |- ran inl = ran (inl |` _V )
29		rnresv 5198	. . . . . . . . 9 |- ran (inr |` _V ) = ran inr
30	29	eqcomi 2234	. . . . . . . 8 |- ran inr = ran (inr |` _V )
31	28, 30	ineq12i 3405	. . . . . . 7 |- ( ran inl i^i ran inr ) = ( ran (inl |` _V ) i^i ran (inr |` _V ) )
32		djuinr 7267	. . . . . . 7 |- ( ran (inl |` _V ) i^i ran (inr |` _V ) ) = (/)
33	31, 32	eqtri 2251	. . . . . 6 |- ( ran inl i^i ran inr ) = (/)
34	33	a1i 9	. . . . 5 |- ( ph -> ( ran inl i^i ran inr ) = (/) )
35	26, 34	sseqtrid 3276	. . . 4 |- ( ph -> ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ (/) )
36		ss0 3534	. . . 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 5373	. . 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 1272	. 2 |- ( ph -> Fun ( ( F o. `'inl ) u. ( G o. `'inr ) ) )
40		df-case 7288	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
41	40	funeqi 5349	. 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 1397   _Vcvv 2801   u. cun 3197   i^i cin 3198   C_ wss 3199   (/)c0 3493   {csn 3670   X. cxp 4725   `'ccnv 4726   dom cdm 4727   ran crn 4728   |` cres 4729   o. ccom 4731   Fun wfun 5322   -->wf 5324   -1-1->wf1 5325   -1-1-onto->wf1o 5327   1oc1o 6580  inlcinl 7249  inrcinr 7250  casecdjucase 7287

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-1st 6308  df-2nd 6309  df-1o 6587  df-inl 7251  df-inr 7252  df-case 7288

This theorem is referenced by:  casef  7292