Theorem djuinj 7304

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 7259	. . . . . . 7 |- (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ A )
2		f1fun 5545	. . . . . . 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 5389	. . . . . 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 5366	. . . . 5 |- ( ( Fun `' `' (inl |` dom R ) /\ Fun `' R ) -> Fun ( `' `' (inl |` dom R ) o. `' R ) )
8	5, 6, 7	sylancr 414	. . . 4 |- ( ph -> Fun ( `' `' (inl |` dom R ) o. `' R ) )
9		cnvco 4915	. . . . 5 |- `' ( R o. `' (inl |` dom R ) ) = ( `' `' (inl |` dom R ) o. `' R )
10	9	funeqi 5347	. . . 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 7260	. . . . . . 7 |- (inr |` dom S ) : dom S -1-1-> ( A ⊔ dom S )
13		f1fun 5545	. . . . . . 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 5389	. . . . . 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 5366	. . . . 5 |- ( ( Fun `' `' (inr |` dom S ) /\ Fun `' S ) -> Fun ( `' `' (inr |` dom S ) o. `' S ) )
19	16, 17, 18	sylancr 414	. . . 4 |- ( ph -> Fun ( `' `' (inr |` dom S ) o. `' S ) )
20		cnvco 4915	. . . . 5 |- `' ( S o. `' (inr |` dom S ) ) = ( `' `' (inr |` dom S ) o. `' S )
21	20	funeqi 5347	. . . 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 4736	. . . . . . 7 |- ran ( R o. `' (inl |` dom R ) ) = dom `' ( R o. `' (inl |` dom R ) )
24		rncoss 5003	. . . . . . 7 |- ran ( R o. `' (inl |` dom R ) ) C_ ran R
25	23, 24	eqsstrri 3260	. . . . . 6 |- dom `' ( R o. `' (inl |` dom R ) ) C_ ran R
26		df-rn 4736	. . . . . . 7 |- ran ( S o. `' (inr |` dom S ) ) = dom `' ( S o. `' (inr |` dom S ) )
27		rncoss 5003	. . . . . . 7 |- ran ( S o. `' (inr |` dom S ) ) C_ ran S
28	26, 27	eqsstrri 3260	. . . . . 6 |- dom `' ( S o. `' (inr |` dom S ) ) C_ ran S
29		ss2in 3435	. . . . . 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 426	. . . . 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 3277	. . . 4 |- ( ph -> ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) C_ (/) )
33		ss0 3535	. . . 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 5371	. . 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 1272	. 2 |- ( ph -> Fun ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) ) )
37		df-djud 7301	. . . . 5 |- ( R ⊔d S ) = ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
38	37	cnveqi 4905	. . . 4 |- `' ( R ⊔d S ) = `' ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
39		cnvun 5142	. . . 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 2252	. . 3 |- `' ( R ⊔d S ) = ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) )
41	40	funeqi 5347	. 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 ) )

Syntax hints:   -> wi 4   = wceq 1397   u. cun 3198   i^i cin 3199   C_ wss 3200   (/)c0 3494   `'ccnv 4724   dom cdm 4725   ran crn 4726   |` cres 4727   o. ccom 4729   Fun wfun 5320   -1-1->wf1 5323   ⊔ cdju 7235  inlcinl 7243  inrcinr 7244   ⊔d cdjud 7300

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dju 7236  df-inl 7245  df-inr 7246  df-djud 7301

This theorem is referenced by: (None)