Theorem casefun 7086

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 7059	. . . . . 6 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
3		f1of1 5462	. . . . . 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 5223	. . . . . 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 5258	. . . 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 7060	. . . . . 6 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
12		f1of1 5462	. . . . . 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 5223	. . . . . 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 5258	. . . 4 |- ( ( Fun G /\ Fun `'inr ) -> Fun ( G o. `'inr ) )
18	10, 16, 17	syl2anc 411	. . 3 |- ( ph -> Fun ( G o. `'inr ) )
19		dmcoss 4898	. . . . . . 7 |- dom ( F o. `'inl ) C_ dom `'inl
20		df-rn 4639	. . . . . . 7 |- ran inl = dom `'inl
21	19, 20	sseqtrri 3192	. . . . . 6 |- dom ( F o. `'inl ) C_ ran inl
22		dmcoss 4898	. . . . . . 7 |- dom ( G o. `'inr ) C_ dom `'inr
23		df-rn 4639	. . . . . . 7 |- ran inr = dom `'inr
24	22, 23	sseqtrri 3192	. . . . . 6 |- dom ( G o. `'inr ) C_ ran inr
25		ss2in 3365	. . . . . 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 5090	. . . . . . . . 9 |- ran (inl |` _V ) = ran inl
28	27	eqcomi 2181	. . . . . . . 8 |- ran inl = ran (inl |` _V )
29		rnresv 5090	. . . . . . . . 9 |- ran (inr |` _V ) = ran inr
30	29	eqcomi 2181	. . . . . . . 8 |- ran inr = ran (inr |` _V )
31	28, 30	ineq12i 3336	. . . . . . 7 |- ( ran inl i^i ran inr ) = ( ran (inl |` _V ) i^i ran (inr |` _V ) )
32		djuinr 7064	. . . . . . 7 |- ( ran (inl |` _V ) i^i ran (inr |` _V ) ) = (/)
33	31, 32	eqtri 2198	. . . . . 6 |- ( ran inl i^i ran inr ) = (/)
34	33	a1i 9	. . . . 5 |- ( ph -> ( ran inl i^i ran inr ) = (/) )
35	26, 34	sseqtrid 3207	. . . 4 |- ( ph -> ( dom ( F o. `'inl ) i^i dom ( G o. `'inr ) ) C_ (/) )
36		ss0 3465	. . . 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 5262	. . 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 1237	. 2 |- ( ph -> Fun ( ( F o. `'inl ) u. ( G o. `'inr ) ) )
40		df-case 7085	. . 3 |- case ( F , G ) = ( ( F o. `'inl ) u. ( G o. `'inr ) )
41	40	funeqi 5239	. 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 1353   _Vcvv 2739   u. cun 3129   i^i cin 3130   C_ wss 3131   (/)c0 3424   {csn 3594   X. cxp 4626   `'ccnv 4627   dom cdm 4628   ran crn 4629   |` cres 4630   o. ccom 4632   Fun wfun 5212   -->wf 5214   -1-1->wf1 5215   -1-1-onto->wf1o 5217   1oc1o 6412  inlcinl 7046  inrcinr 7047  casecdjucase 7084

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-inl 7048  df-inr 7049  df-case 7085

This theorem is referenced by:  casef  7089