Theorem caseinj 7054

Description: The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
caseinj.r	|- ( ph -> Fun `' R )
caseinj.s	|- ( ph -> Fun `' S )
caseinj.disj	|- ( ph -> ( ran R i^i ran S ) = (/) )

Assertion

Ref	Expression
caseinj	|- ( ph -> Fun `'case ( R , S ) )

Proof of Theorem caseinj

Step	Hyp	Ref	Expression
1		df-inl 7012	. . . . . . 7 |- inl = ( y e. _V |-> <. (/) , y >. )
2	1	funmpt2 5227	. . . . . 6 |- Fun inl
3		funcnvcnv 5247	. . . . . 6 |- ( Fun inl -> Fun `' `'inl )
4	2, 3	ax-mp 5	. . . . 5 |- Fun `' `'inl
5		caseinj.r	. . . . 5 |- ( ph -> Fun `' R )
6		funco 5228	. . . . 5 |- ( ( Fun `' `'inl /\ Fun `' R ) -> Fun ( `' `'inl o. `' R ) )
7	4, 5, 6	sylancr 411	. . . 4 |- ( ph -> Fun ( `' `'inl o. `' R ) )
8		cnvco 4789	. . . . 5 |- `' ( R o. `'inl ) = ( `' `'inl o. `' R )
9	8	funeqi 5209	. . . 4 |- ( Fun `' ( R o. `'inl ) <-> Fun ( `' `'inl o. `' R ) )
10	7, 9	sylibr 133	. . 3 |- ( ph -> Fun `' ( R o. `'inl ) )
11		df-inr 7013	. . . . . . 7 |- inr = ( x e. _V |-> <. 1o , x >. )
12	11	funmpt2 5227	. . . . . 6 |- Fun inr
13		funcnvcnv 5247	. . . . . 6 |- ( Fun inr -> Fun `' `'inr )
14	12, 13	ax-mp 5	. . . . 5 |- Fun `' `'inr
15		caseinj.s	. . . . 5 |- ( ph -> Fun `' S )
16		funco 5228	. . . . 5 |- ( ( Fun `' `'inr /\ Fun `' S ) -> Fun ( `' `'inr o. `' S ) )
17	14, 15, 16	sylancr 411	. . . 4 |- ( ph -> Fun ( `' `'inr o. `' S ) )
18		cnvco 4789	. . . . 5 |- `' ( S o. `'inr ) = ( `' `'inr o. `' S )
19	18	funeqi 5209	. . . 4 |- ( Fun `' ( S o. `'inr ) <-> Fun ( `' `'inr o. `' S ) )
20	17, 19	sylibr 133	. . 3 |- ( ph -> Fun `' ( S o. `'inr ) )
21		df-rn 4615	. . . . . . 7 |- ran ( R o. `'inl ) = dom `' ( R o. `'inl )
22		rncoss 4874	. . . . . . 7 |- ran ( R o. `'inl ) C_ ran R
23	21, 22	eqsstrri 3175	. . . . . 6 |- dom `' ( R o. `'inl ) C_ ran R
24		df-rn 4615	. . . . . . 7 |- ran ( S o. `'inr ) = dom `' ( S o. `'inr )
25		rncoss 4874	. . . . . . 7 |- ran ( S o. `'inr ) C_ ran S
26	24, 25	eqsstrri 3175	. . . . . 6 |- dom `' ( S o. `'inr ) C_ ran S
27		ss2in 3350	. . . . . 6 |- ( ( dom `' ( R o. `'inl ) C_ ran R /\ dom `' ( S o. `'inr ) C_ ran S ) -> ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) C_ ( ran R i^i ran S ) )
28	23, 26, 27	mp2an 423	. . . . 5 |- ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) C_ ( ran R i^i ran S )
29		caseinj.disj	. . . . 5 |- ( ph -> ( ran R i^i ran S ) = (/) )
30	28, 29	sseqtrid 3192	. . . 4 |- ( ph -> ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) C_ (/) )
31		ss0 3449	. . . 4 |- ( ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) C_ (/) -> ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) = (/) )
32	30, 31	syl 14	. . 3 |- ( ph -> ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) = (/) )
33		funun 5232	. . 3 |- ( ( ( Fun `' ( R o. `'inl ) /\ Fun `' ( S o. `'inr ) ) /\ ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) = (/) ) -> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
34	10, 20, 32, 33	syl21anc 1227	. 2 |- ( ph -> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
35		df-case 7049	. . . . 5 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
36	35	cnveqi 4779	. . . 4 |- `'case ( R , S ) = `' ( ( R o. `'inl ) u. ( S o. `'inr ) )
37		cnvun 5009	. . . 4 |- `' ( ( R o. `'inl ) u. ( S o. `'inr ) ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
38	36, 37	eqtri 2186	. . 3 |- `'case ( R , S ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
39	38	funeqi 5209	. 2 |- ( Fun `'case ( R , S ) <-> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
40	34, 39	sylibr 133	1 |- ( ph -> Fun `'case ( R , S ) )

Syntax hints:   -> wi 4   = wceq 1343   _Vcvv 2726   u. cun 3114   i^i cin 3115   C_ wss 3116   (/)c0 3409   <.cop 3579   `'ccnv 4603   dom cdm 4604   ran crn 4605   o. ccom 4608   Fun wfun 5182   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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  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-ral 2449  df-rex 2450  df-v 2728  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-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-inl 7012  df-inr 7013  df-case 7049

This theorem is referenced by:  casef1  7055