Theorem caseinj 6974

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 6932	. . . . . . 7 |- inl = ( y e. _V |-> <. (/) , y >. )
2	1	funmpt2 5162	. . . . . 6 |- Fun inl
3		funcnvcnv 5182	. . . . . 6 |- ( Fun inl -> Fun `' `'inl )
4	2, 3	ax-mp 5	. . . . 5 |- Fun `' `'inl
5		caseinj.r	. . . . 5 |- ( ph -> Fun `' R )
6		funco 5163	. . . . 5 |- ( ( Fun `' `'inl /\ Fun `' R ) -> Fun ( `' `'inl o. `' R ) )
7	4, 5, 6	sylancr 410	. . . 4 |- ( ph -> Fun ( `' `'inl o. `' R ) )
8		cnvco 4724	. . . . 5 |- `' ( R o. `'inl ) = ( `' `'inl o. `' R )
9	8	funeqi 5144	. . . 4 |- ( Fun `' ( R o. `'inl ) <-> Fun ( `' `'inl o. `' R ) )
10	7, 9	sylibr 133	. . 3 |- ( ph -> Fun `' ( R o. `'inl ) )
11		df-inr 6933	. . . . . . 7 |- inr = ( x e. _V |-> <. 1o , x >. )
12	11	funmpt2 5162	. . . . . 6 |- Fun inr
13		funcnvcnv 5182	. . . . . 6 |- ( Fun inr -> Fun `' `'inr )
14	12, 13	ax-mp 5	. . . . 5 |- Fun `' `'inr
15		caseinj.s	. . . . 5 |- ( ph -> Fun `' S )
16		funco 5163	. . . . 5 |- ( ( Fun `' `'inr /\ Fun `' S ) -> Fun ( `' `'inr o. `' S ) )
17	14, 15, 16	sylancr 410	. . . 4 |- ( ph -> Fun ( `' `'inr o. `' S ) )
18		cnvco 4724	. . . . 5 |- `' ( S o. `'inr ) = ( `' `'inr o. `' S )
19	18	funeqi 5144	. . . 4 |- ( Fun `' ( S o. `'inr ) <-> Fun ( `' `'inr o. `' S ) )
20	17, 19	sylibr 133	. . 3 |- ( ph -> Fun `' ( S o. `'inr ) )
21		df-rn 4550	. . . . . . 7 |- ran ( R o. `'inl ) = dom `' ( R o. `'inl )
22		rncoss 4809	. . . . . . 7 |- ran ( R o. `'inl ) C_ ran R
23	21, 22	eqsstrri 3130	. . . . . 6 |- dom `' ( R o. `'inl ) C_ ran R
24		df-rn 4550	. . . . . . 7 |- ran ( S o. `'inr ) = dom `' ( S o. `'inr )
25		rncoss 4809	. . . . . . 7 |- ran ( S o. `'inr ) C_ ran S
26	24, 25	eqsstrri 3130	. . . . . 6 |- dom `' ( S o. `'inr ) C_ ran S
27		ss2in 3304	. . . . . 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 422	. . . . 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 3147	. . . 4 |- ( ph -> ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) C_ (/) )
31		ss0 3403	. . . 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 5167	. . 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 1215	. 2 |- ( ph -> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
35		df-case 6969	. . . . 5 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
36	35	cnveqi 4714	. . . 4 |- `'case ( R , S ) = `' ( ( R o. `'inl ) u. ( S o. `'inr ) )
37		cnvun 4944	. . . 4 |- `' ( ( R o. `'inl ) u. ( S o. `'inr ) ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
38	36, 37	eqtri 2160	. . 3 |- `'case ( R , S ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
39	38	funeqi 5144	. 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 1331   _Vcvv 2686   u. cun 3069   i^i cin 3070   C_ wss 3071   (/)c0 3363   <.cop 3530   `'ccnv 4538   dom cdm 4539   ran crn 4540   o. ccom 4543   Fun wfun 5117   1oc1o 6306  inlcinl 6930  inrcinr 6931  casecdjucase 6968

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-inl 6932  df-inr 6933  df-case 6969

This theorem is referenced by:  casef1  6975