Theorem djuinj 7452

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 7407	. . . . . . 7 |- (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ A )
2		f1fun 5604	. . . . . . 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 5443	. . . . . 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 5420	. . . . 5 |- ( ( Fun `' `' (inl |` dom R ) /\ Fun `' R ) -> Fun ( `' `' (inl |` dom R ) o. `' R ) )
8	5, 6, 7	sylancr 418	. . . 4 |- ( ph -> Fun ( `' `' (inl |` dom R ) o. `' R ) )
9		cnvco 4967	. . . . 5 |- `' ( R o. `' (inl |` dom R ) ) = ( `' `' (inl |` dom R ) o. `' R )
10	9	funeqi 5401	. . . 4 |- ( Fun `' ( R o. `' (inl |` dom R ) ) <-> Fun ( `' `' (inl |` dom R ) o. `' R ) )
11	8, 10	sylibr 134	. . 3 |- ( ph -> Fun `' ( R o. `' (inl |` dom R ) ) )
12		inrresf1 7408	. . . . . . 7 |- (inr |` dom S ) : dom S -1-1-> ( A ⊔ dom S )
13		f1fun 5604	. . . . . . 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 5443	. . . . . 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 5420	. . . . 5 |- ( ( Fun `' `' (inr |` dom S ) /\ Fun `' S ) -> Fun ( `' `' (inr |` dom S ) o. `' S ) )
19	16, 17, 18	sylancr 418	. . . 4 |- ( ph -> Fun ( `' `' (inr |` dom S ) o. `' S ) )
20		cnvco 4967	. . . . 5 |- `' ( S o. `' (inr |` dom S ) ) = ( `' `' (inr |` dom S ) o. `' S )
21	20	funeqi 5401	. . . 4 |- ( Fun `' ( S o. `' (inr |` dom S ) ) <-> Fun ( `' `' (inr |` dom S ) o. `' S ) )
22	19, 21	sylibr 134	. . 3 |- ( ph -> Fun `' ( S o. `' (inr |` dom S ) ) )
23		df-rn 4787	. . . . . . 7 |- ran ( R o. `' (inl |` dom R ) ) = dom `' ( R o. `' (inl |` dom R ) )
24		rncoss 5055	. . . . . . 7 |- ran ( R o. `' (inl |` dom R ) ) C_ ran R
25	23, 24	eqsstrri 3281	. . . . . 6 |- dom `' ( R o. `' (inl |` dom R ) ) C_ ran R
26		df-rn 4787	. . . . . . 7 |- ran ( S o. `' (inr |` dom S ) ) = dom `' ( S o. `' (inr |` dom S ) )
27		rncoss 5055	. . . . . . 7 |- ran ( S o. `' (inr |` dom S ) ) C_ ran S
28	26, 27	eqsstrri 3281	. . . . . 6 |- dom `' ( S o. `' (inr |` dom S ) ) C_ ran S
29		ss2in 3459	. . . . . 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 430	. . . . 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 3298	. . . 4 |- ( ph -> ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) C_ (/) )
33		ss0 3563	. . . 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 5425	. . 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 1277	. 2 |- ( ph -> Fun ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) ) )
37		df-djud 7449	. . . . 5 |- ( R ⊔d S ) = ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
38	37	cnveqi 4957	. . . 4 |- `' ( R ⊔d S ) = `' ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
39		cnvun 5196	. . . 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 2259	. . 3 |- `' ( R ⊔d S ) = ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) )
41	40	funeqi 5401	. 2 |- ( Fun `' ( R ⊔d S ) <-> Fun ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) ) )
42	36, 41	sylibr 134	1 |- ( ph -> Fun `' ( R ⊔d S ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   `'ccnv 4775   dom cdm 4776   ran crn 4777   |` cres 4778   o. ccom 4780   Fun wfun 5374   -1-1->wf1 5377   ⊔ cdju 7383  inlcinl 7391  inrcinr 7392   ⊔d cdjud 7448

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-dju 7384  df-inl 7393  df-inr 7394  df-djud 7449

This theorem is used by: (None)