Theorem djuinj 7234

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 7189	. . . . . . 7 |- (inl |` dom R ) : dom R -1-1-> ( dom R ⊔ A )
2		f1fun 5506	. . . . . . 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 5352	. . . . . 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 5330	. . . . 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 4881	. . . . 5 |- `' ( R o. `' (inl |` dom R ) ) = ( `' `' (inl |` dom R ) o. `' R )
10	9	funeqi 5311	. . . 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 7190	. . . . . . 7 |- (inr |` dom S ) : dom S -1-1-> ( A ⊔ dom S )
13		f1fun 5506	. . . . . . 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 5352	. . . . . 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 5330	. . . . 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 4881	. . . . 5 |- `' ( S o. `' (inr |` dom S ) ) = ( `' `' (inr |` dom S ) o. `' S )
21	20	funeqi 5311	. . . 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 4704	. . . . . . 7 |- ran ( R o. `' (inl |` dom R ) ) = dom `' ( R o. `' (inl |` dom R ) )
24		rncoss 4968	. . . . . . 7 |- ran ( R o. `' (inl |` dom R ) ) C_ ran R
25	23, 24	eqsstrri 3234	. . . . . 6 |- dom `' ( R o. `' (inl |` dom R ) ) C_ ran R
26		df-rn 4704	. . . . . . 7 |- ran ( S o. `' (inr |` dom S ) ) = dom `' ( S o. `' (inr |` dom S ) )
27		rncoss 4968	. . . . . . 7 |- ran ( S o. `' (inr |` dom S ) ) C_ ran S
28	26, 27	eqsstrri 3234	. . . . . 6 |- dom `' ( S o. `' (inr |` dom S ) ) C_ ran S
29		ss2in 3409	. . . . . 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 3251	. . . 4 |- ( ph -> ( dom `' ( R o. `' (inl |` dom R ) ) i^i dom `' ( S o. `' (inr |` dom S ) ) ) C_ (/) )
33		ss0 3509	. . . 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 5334	. . 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 1249	. 2 |- ( ph -> Fun ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) ) )
37		df-djud 7231	. . . . 5 |- ( R ⊔d S ) = ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
38	37	cnveqi 4871	. . . 4 |- `' ( R ⊔d S ) = `' ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
39		cnvun 5107	. . . 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 2228	. . 3 |- `' ( R ⊔d S ) = ( `' ( R o. `' (inl |` dom R ) ) u. `' ( S o. `' (inr |` dom S ) ) )
41	40	funeqi 5311	. 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 1373   u. cun 3172   i^i cin 3173   C_ wss 3174   (/)c0 3468   `'ccnv 4692   dom cdm 4693   ran crn 4694   |` cres 4695   o. ccom 4697   Fun wfun 5284   -1-1->wf1 5287   ⊔ cdju 7165  inlcinl 7173  inrcinr 7174   ⊔d cdjud 7230

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-1o 6525  df-dju 7166  df-inl 7175  df-inr 7176  df-djud 7231

This theorem is referenced by: (None)