Theorem djuinj 7083

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

Hypotheses

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

Assertion

Ref	Expression
djuinj	|- ( ph -> Fun `' ( R ⊔d S ) )

Proof of Theorem djuinj

Step	Hyp	Ref	Expression
1		inlresf1 7038	. . . . . . 7 |- (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ A )
2		f1fun 5406	. . . . . . 7 |- ( (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ A ) -> Fun (inl |` dom R ) )
3	1, 2	ax-mp 5	. . . . . 6 |- Fun (inl |` dom R )
4		funcnvcnv 5257	. . . . . 6 |- ( Fun (inl |` dom R ) -> Fun `' `' (inl |` dom R ) )
5	3, 4	ax-mp 5	. . . . 5 |- Fun `' `' (inl |` dom R )
6		djuinj.r	. . . . 5 |- ( ph -> Fun `' R )
7		funco 5238	. . . . 5 |- ( ( Fun `' `' (inl |` dom R ) /\ Fun `' R ) -> Fun ( `' `' (inl |` dom R ) o. `' R ) )
8	5, 6, 7	sylancr 412	. . . 4 |- ( ph -> Fun ( `' `' (inl |` dom R ) o. `' R ) )
9		cnvco 4796	. . . . 5 |- `' ( R o. `' (inl |` dom R ) ) = ( `' `' (inl |` dom R ) o. `' R )
10	9	funeqi 5219	. . . 4 |- ( Fun `' ( R o. `' (inl |` dom R ) ) <-> Fun ( `' `' (inl |` dom R ) o. `' R ) )
11	8, 10	sylibr 133	. . 3 |- ( ph -> Fun `' ( R o. `' (inl |` dom R ) ) )
12		inrresf1 7039	. . . . . . 7 |- (inr |` dom S ) : dom S -1-1-> ( A ⊔ dom S )
13		f1fun 5406	. . . . . . 7 |- ( (inr |` dom S ) : dom S -1-1-> ( A ⊔ dom S ) -> Fun (inr |` dom S ) )
14	12, 13	ax-mp 5	. . . . . 6 |- Fun (inr |` dom S )
15		funcnvcnv 5257	. . . . . 6 |- ( Fun (inr |` dom S ) -> Fun `' `' (inr |` dom S ) )
16	14, 15	ax-mp 5	. . . . 5 |- Fun `' `' (inr |` dom S )
17		djuinj.s	. . . . 5 |- ( ph -> Fun `' S )
18		funco 5238	. . . . 5 |- ( ( Fun `' `' (inr |` dom S ) /\ Fun `' S ) -> Fun ( `' `' (inr |` dom S ) o. `' S ) )
19	16, 17, 18	sylancr 412	. . . 4 |- ( ph -> Fun ( `' `' (inr |` dom S ) o. `' S ) )
20		cnvco 4796	. . . . 5 |- `' ( S o. `' (inr |` dom S ) ) = ( `' `' (inr |` dom S ) o. `' S )
21	20	funeqi 5219	. . . 4 |- ( Fun `' ( S o. `' (inr |` dom S ) ) <-> Fun ( `' `' (inr |` dom S ) o. `' S ) )
22	19, 21	sylibr 133	. . 3 |- ( ph -> Fun `' ( S o. `' (inr |` dom S ) ) )
23		df-rn 4622	. . . . . . 7 |- ran ( R o. `' (inl |` dom R ) ) = dom `' ( R o. `' (inl |` dom R ) )
24		rncoss 4881	. . . . . . 7 |- ran ( R o. `' (inl |` dom R ) ) C_ ran R
25	23, 24	eqsstrri 3180	. . . . . 6 |- dom `' ( R o. `' (inl |` dom R ) ) C_ ran R
26		df-rn 4622	. . . . . . 7 |- ran ( S o. `' (inr |` dom S ) ) = dom `' ( S o. `' (inr |` dom S ) )
27		rncoss 4881	. . . . . . 7 |- ran ( S o. `' (inr |` dom S ) ) C_ ran S
28	26, 27	eqsstrri 3180	. . . . . 6 |- dom `' ( S o. `' (inr |` dom S ) ) C_ ran S
29		ss2in 3355	. . . . . 6 |- ( ( dom `' ( R o. `' (inl |` dom R ) ) C_ ran R /\ dom `' ( S o. `' (inr |` dom S ) ) C_ ran S ) -> ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) C_ ( ran R i^i ran S ) )
30	25, 28, 29	mp2an 424	. . . . 5 |- ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) C_ ( ran R i^i ran S )
31		djuinj.disj	. . . . 5 |- ( ph -> ( ran R i^i ran S ) = (/) )
32	30, 31	sseqtrid 3197	. . . 4 |- ( ph -> ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) C_ (/) )
33		ss0 3455	. . . 4 |- ( ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) C_ (/) -> ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) = (/) )
34	32, 33	syl 14	. . 3 |- ( ph -> ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) = (/) )
35		funun 5242	. . 3 |- ( ( ( Fun `' ( R o. `' (inl |` dom R ) ) /\ Fun `' ( S o. `' (inr |` dom S ) ) ) /\ ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) = (/) ) -> Fun ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) ) )
36	11, 22, 34, 35	syl21anc 1232	. 2 |- ( ph -> Fun ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) ) )
37		df-djud 7080	. . . . 5 |- ( R ⊔d S ) = ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
38	37	cnveqi 4786	. . . 4 |- `' ( R ⊔d S ) = `' ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
39		cnvun 5016	. . . 4 |- `' ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) ) = ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) )
40	38, 39	eqtri 2191	. . 3 |- `' ( R ⊔d S ) = ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) )
41	40	funeqi 5219	. 2 |- ( Fun `' ( R ⊔d S ) <-> Fun ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) ) )
42	36, 41	sylibr 133	1 |- ( ph -> Fun `' ( R ⊔d S ) )

Syntax hints:   -> wi 4   = wceq 1348   u. cun 3119   i^i cin 3120   C_ wss 3121   (/)c0 3414   `'ccnv 4610   dom cdm 4611   ran crn 4612   |` cres 4613   o. ccom 4615   Fun wfun 5192   -1-1->wf1 5195   ⊔ cdju 7014  inlcinl 7022  inrcinr 7023   ⊔d cdjud 7079

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025  df-djud 7080

This theorem is referenced by: (None)