Theorem caseinj 6778

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 6737	. . . . . . 7 |- inl = ( y e. _V |-> <. (/) , y >. )
2	1	funmpt2 5053	. . . . . 6 |- Fun inl
3		funcnvcnv 5073	. . . . . 6 |- ( Fun inl -> Fun `' `'inl )
4	2, 3	ax-mp 7	. . . . 5 |- Fun `' `'inl
5		caseinj.r	. . . . 5 |- ( ph -> Fun `' R )
6		funco 5054	. . . . 5 |- ( ( Fun `' `'inl /\ Fun `' R ) -> Fun ( `' `'inl o. `' R ) )
7	4, 5, 6	sylancr 405	. . . 4 |- ( ph -> Fun ( `' `'inl o. `' R ) )
8		cnvco 4621	. . . . 5 |- `' ( R o. `'inl ) = ( `' `'inl o. `' R )
9	8	funeqi 5036	. . . 4 |- ( Fun `' ( R o. `'inl ) <-> Fun ( `' `'inl o. `' R ) )
10	7, 9	sylibr 132	. . 3 |- ( ph -> Fun `' ( R o. `'inl ) )
11		df-inr 6738	. . . . . . 7 |- inr = ( x e. _V |-> <. 1o , x >. )
12	11	funmpt2 5053	. . . . . 6 |- Fun inr
13		funcnvcnv 5073	. . . . . 6 |- ( Fun inr -> Fun `' `'inr )
14	12, 13	ax-mp 7	. . . . 5 |- Fun `' `'inr
15		caseinj.s	. . . . 5 |- ( ph -> Fun `' S )
16		funco 5054	. . . . 5 |- ( ( Fun `' `'inr /\ Fun `' S ) -> Fun ( `' `'inr o. `' S ) )
17	14, 15, 16	sylancr 405	. . . 4 |- ( ph -> Fun ( `' `'inr o. `' S ) )
18		cnvco 4621	. . . . 5 |- `' ( S o. `'inr ) = ( `' `'inr o. `' S )
19	18	funeqi 5036	. . . 4 |- ( Fun `' ( S o. `'inr ) <-> Fun ( `' `'inr o. `' S ) )
20	17, 19	sylibr 132	. . 3 |- ( ph -> Fun `' ( S o. `'inr ) )
21		df-rn 4449	. . . . . . 7 |- ran ( R o. `'inl ) = dom `' ( R o. `'inl )
22		rncoss 4703	. . . . . . 7 |- ran ( R o. `'inl ) C_ ran R
23	21, 22	eqsstr3i 3057	. . . . . 6 |- dom `' ( R o. `'inl ) C_ ran R
24		df-rn 4449	. . . . . . 7 |- ran ( S o. `'inr ) = dom `' ( S o. `'inr )
25		rncoss 4703	. . . . . . 7 |- ran ( S o. `'inr ) C_ ran S
26	24, 25	eqsstr3i 3057	. . . . . 6 |- dom `' ( S o. `'inr ) C_ ran S
27		ss2in 3227	. . . . . 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 417	. . . . 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	syl5sseq 3074	. . . 4 |- ( ph -> ( dom `' ( R o. `'inl ) i^i dom `' ( S o. `'inr ) ) C_ (/) )
31		ss0 3323	. . . 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 5058	. . 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 1173	. 2 |- ( ph -> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
35		df-case 6773	. . . . 5 |- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
36	35	cnveqi 4611	. . . 4 |- `'case ( R , S ) = `' ( ( R o. `'inl ) u. ( S o. `'inr ) )
37		cnvun 4837	. . . 4 |- `' ( ( R o. `'inl ) u. ( S o. `'inr ) ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
38	36, 37	eqtri 2108	. . 3 |- `'case ( R , S ) = ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) )
39	38	funeqi 5036	. 2 |- ( Fun `'case ( R , S ) <-> Fun ( `' ( R o. `'inl ) u. `' ( S o. `'inr ) ) )
40	34, 39	sylibr 132	1 |- ( ph -> Fun `'case ( R , S ) )

Syntax hints:   -> wi 4   = wceq 1289   _Vcvv 2619   u. cun 2997   i^i cin 2998   C_ wss 2999   (/)c0 3286   <.cop 3449   `'ccnv 4437   dom cdm 4438   ran crn 4439   o. ccom 4442   Fun wfun 5009   1oc1o 6174  inlcinl 6735  inrcinr 6736  casecdjucase 6772

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-fun 5017  df-inl 6737  df-inr 6738  df-case 6773

This theorem is referenced by:  casef1  6779