Theorem caseinj 7036

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 6994	. . . . . . 7 |- inl = ( y e. _V |-> <. (/) , y >. )
2	1	funmpt2 5212	. . . . . 6 |- Fun inl
3		funcnvcnv 5232	. . . . . 6 |- ( Fun inl -> Fun `' `'inl )
4	2, 3	ax-mp 5	. . . . 5 |- Fun `' `'inl
5		caseinj.r	. . . . 5 |- ( ph -> Fun `' R )
6		funco 5213	. . . . 5 |- ( ( Fun `' `'inl /\ Fun `' R ) -> Fun ( `' `'inl o. `' R ) )
7	4, 5, 6	sylancr 411	. . . 4 |- ( ph -> Fun ( `' `'inl o. `' R ) )
8		cnvco 4774	. . . . 5 |- `' ( R o. `'inl ) = ( `' `'inl o. `' R )
9	8	funeqi 5194	. . . 4 |- ( Fun `' ( R o. `'inl ) <-> Fun ( `' `'inl o. `' R ) )
10	7, 9	sylibr 133	. . 3 |- ( ph -> Fun `' ( R o. `'inl ) )
11		df-inr 6995	. . . . . . 7 |- inr = ( x e. _V |-> <. 1o , x >. )
12	11	funmpt2 5212	. . . . . 6 |- Fun inr
13		funcnvcnv 5232	. . . . . 6 |- ( Fun inr -> Fun `' `'inr )
14	12, 13	ax-mp 5	. . . . 5 |- Fun `' `'inr
15		caseinj.s	. . . . 5 |- ( ph -> Fun `' S )
16		funco 5213	. . . . 5 |- ( ( Fun `' `'inr /\ Fun `' S ) -> Fun ( `' `'inr o. `' S ) )
17	14, 15, 16	sylancr 411	. . . 4 |- ( ph -> Fun ( `' `'inr o. `' S ) )
18		cnvco 4774	. . . . 5 |- `' ( S o. `'inr ) = ( `' `'inr o. `' S )
19	18	funeqi 5194	. . . 4 |- ( Fun `' ( S o. `'inr ) <-> Fun ( `' `'inr o. `' S ) )
20	17, 19	sylibr 133	. . 3 |- ( ph -> Fun `' ( S o. `'inr ) )
21		df-rn 4600	. . . . . . 7 |- ran ( R o. `'inl ) = dom `' ( R o. `'inl )
22		rncoss 4859	. . . . . . 7 |- ran ( R o. `'inl ) C_ ran R
23	21, 22	eqsstrri 3161	. . . . . 6 |- dom `' ( R o. `'inl ) C_ ran R
24		df-rn 4600	. . . . . . 7 |- ran ( S o. `'inr ) = dom `' ( S o. `'inr )
25		rncoss 4859	. . . . . . 7 |- ran ( S o. `'inr ) C_ ran S
26	24, 25	eqsstrri 3161	. . . . . 6 |- dom `' ( S o. `'inr ) C_ ran S
27		ss2in 3336	. . . . . 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 3178	. . . 4 |- ( ph -> ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) C_ (/) )
31		ss0 3435	. . . 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 5217	. . 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 1219	. 2 |- ( ph -> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
35		df-case 7031	. . . . 5 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
36	35	cnveqi 4764	. . . 4 |- `'case ( R , S ) = `' ( ( R o. `'inl ) u. ( S o. `'inr ) )
37		cnvun 4994	. . . 4 |- `' ( ( R o. `'inl ) u. ( S o. `'inr ) ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
38	36, 37	eqtri 2178	. . 3 |- `'case ( R , S ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
39	38	funeqi 5194	. 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 1335   _Vcvv 2712   u. cun 3100   i^i cin 3101   C_ wss 3102   (/)c0 3395   <.cop 3564   `'ccnv 4588   dom cdm 4589   ran crn 4590   o. ccom 4593   Fun wfun 5167   1oc1o 6359  inlcinl 6992  inrcinr 6993  casecdjucase 7030

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4029  df-mpt 4030  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-fun 5175  df-inl 6994  df-inr 6995  df-case 7031

This theorem is referenced by:  casef1  7037