Theorem caseinj 7090

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 7048	. . . . . . 7 |- inl = ( y e. _V |-> <. (/) , y >. )
2	1	funmpt2 5257	. . . . . 6 |- Fun inl
3		funcnvcnv 5277	. . . . . 6 |- ( Fun inl -> Fun `' `'inl )
4	2, 3	ax-mp 5	. . . . 5 |- Fun `' `'inl
5		caseinj.r	. . . . 5 |- ( ph -> Fun `' R )
6		funco 5258	. . . . 5 |- ( ( Fun `' `'inl /\ Fun `' R ) -> Fun ( `' `'inl o. `' R ) )
7	4, 5, 6	sylancr 414	. . . 4 |- ( ph -> Fun ( `' `'inl o. `' R ) )
8		cnvco 4814	. . . . 5 |- `' ( R o. `'inl ) = ( `' `'inl o. `' R )
9	8	funeqi 5239	. . . 4 |- ( Fun `' ( R o. `'inl ) <-> Fun ( `' `'inl o. `' R ) )
10	7, 9	sylibr 134	. . 3 |- ( ph -> Fun `' ( R o. `'inl ) )
11		df-inr 7049	. . . . . . 7 |- inr = ( x e. _V |-> <. 1o , x >. )
12	11	funmpt2 5257	. . . . . 6 |- Fun inr
13		funcnvcnv 5277	. . . . . 6 |- ( Fun inr -> Fun `' `'inr )
14	12, 13	ax-mp 5	. . . . 5 |- Fun `' `'inr
15		caseinj.s	. . . . 5 |- ( ph -> Fun `' S )
16		funco 5258	. . . . 5 |- ( ( Fun `' `'inr /\ Fun `' S ) -> Fun ( `' `'inr o. `' S ) )
17	14, 15, 16	sylancr 414	. . . 4 |- ( ph -> Fun ( `' `'inr o. `' S ) )
18		cnvco 4814	. . . . 5 |- `' ( S o. `'inr ) = ( `' `'inr o. `' S )
19	18	funeqi 5239	. . . 4 |- ( Fun `' ( S o. `'inr ) <-> Fun ( `' `'inr o. `' S ) )
20	17, 19	sylibr 134	. . 3 |- ( ph -> Fun `' ( S o. `'inr ) )
21		df-rn 4639	. . . . . . 7 |- ran ( R o. `'inl ) = dom `' ( R o. `'inl )
22		rncoss 4899	. . . . . . 7 |- ran ( R o. `'inl ) C_ ran R
23	21, 22	eqsstrri 3190	. . . . . 6 |- dom `' ( R o. `'inl ) C_ ran R
24		df-rn 4639	. . . . . . 7 |- ran ( S o. `'inr ) = dom `' ( S o. `'inr )
25		rncoss 4899	. . . . . . 7 |- ran ( S o. `'inr ) C_ ran S
26	24, 25	eqsstrri 3190	. . . . . 6 |- dom `' ( S o. `'inr ) C_ ran S
27		ss2in 3365	. . . . . 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 426	. . . . 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 3207	. . . 4 |- ( ph -> ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) C_ (/) )
31		ss0 3465	. . . 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 5262	. . 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 1237	. 2 |- ( ph -> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
35		df-case 7085	. . . . 5 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
36	35	cnveqi 4804	. . . 4 |- `'case ( R , S ) = `' ( ( R o. `'inl ) u. ( S o. `'inr ) )
37		cnvun 5036	. . . 4 |- `' ( ( R o. `'inl ) u. ( S o. `'inr ) ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
38	36, 37	eqtri 2198	. . 3 |- `'case ( R , S ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
39	38	funeqi 5239	. 2 |- ( Fun `'case ( R , S ) <-> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
40	34, 39	sylibr 134	1 |- ( ph -> Fun `'case ( R , S ) )

Syntax hints:   -> wi 4   = wceq 1353   _Vcvv 2739   u. cun 3129   i^i cin 3130   C_ wss 3131   (/)c0 3424   <.cop 3597   `'ccnv 4627   dom cdm 4628   ran crn 4629   o. ccom 4632   Fun wfun 5212   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-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2741  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-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-inl 7048  df-inr 7049  df-case 7085

This theorem is referenced by:  casef1  7091