Theorem djuinj 7123

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	⊢ (𝜑 → Fun ◡𝑅)
djuinj.s	⊢ (𝜑 → Fun ◡𝑆)
djuinj.disj	⊢ (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)

Assertion

Ref	Expression
djuinj	⊢ (𝜑 → Fun ◡(𝑅 ⊔d 𝑆))

Proof of Theorem djuinj

Step	Hyp	Ref	Expression
1		inlresf1 7078	. . . . . . 7 ⊢ (inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ 𝐴)
2		f1fun 5439	. . . . . . 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 5290	. . . . . 6 ⊢ (Fun (inl ↾ dom 𝑅) → Fun ◡◡(inl ↾ dom 𝑅))
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun ◡◡(inl ↾ dom 𝑅)
6		djuinj.r	. . . . 5 ⊢ (𝜑 → Fun ◡𝑅)
7		funco 5271	. . . . 5 ⊢ ((Fun ◡◡(inl ↾ dom 𝑅) ∧ Fun ◡𝑅) → Fun (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅))
8	5, 6, 7	sylancr 414	. . . 4 ⊢ (𝜑 → Fun (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅))
9		cnvco 4827	. . . . 5 ⊢ ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) = (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅)
10	9	funeqi 5252	. . . 4 ⊢ (Fun ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ↔ Fun (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅))
11	8, 10	sylibr 134	. . 3 ⊢ (𝜑 → Fun ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)))
12		inrresf1 7079	. . . . . . 7 ⊢ (inr ↾ dom 𝑆):dom 𝑆–1-1→(𝐴 ⊔ dom 𝑆)
13		f1fun 5439	. . . . . . 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 5290	. . . . . 6 ⊢ (Fun (inr ↾ dom 𝑆) → Fun ◡◡(inr ↾ dom 𝑆))
16	14, 15	ax-mp 5	. . . . 5 ⊢ Fun ◡◡(inr ↾ dom 𝑆)
17		djuinj.s	. . . . 5 ⊢ (𝜑 → Fun ◡𝑆)
18		funco 5271	. . . . 5 ⊢ ((Fun ◡◡(inr ↾ dom 𝑆) ∧ Fun ◡𝑆) → Fun (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆))
19	16, 17, 18	sylancr 414	. . . 4 ⊢ (𝜑 → Fun (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆))
20		cnvco 4827	. . . . 5 ⊢ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) = (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆)
21	20	funeqi 5252	. . . 4 ⊢ (Fun ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) ↔ Fun (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆))
22	19, 21	sylibr 134	. . 3 ⊢ (𝜑 → Fun ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
23		df-rn 4652	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡(inl ↾ dom 𝑅)) = dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅))
24		rncoss 4912	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅
25	23, 24	eqsstrri 3203	. . . . . 6 ⊢ dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅
26		df-rn 4652	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡(inr ↾ dom 𝑆)) = dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))
27		rncoss 4912	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆
28	26, 27	eqsstrri 3203	. . . . . 6 ⊢ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆
29		ss2in 3378	. . . . . 6 ⊢ ((dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅 ∧ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆) → (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) ⊆ (ran 𝑅 ∩ ran 𝑆))
30	25, 28, 29	mp2an 426	. . . . 5 ⊢ (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) ⊆ (ran 𝑅 ∩ ran 𝑆)
31		djuinj.disj	. . . . 5 ⊢ (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)
32	30, 31	sseqtrid 3220	. . . 4 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) ⊆ ∅)
33		ss0 3478	. . . 4 ⊢ ((dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) ⊆ ∅ → (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) = ∅)
34	32, 33	syl 14	. . 3 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) = ∅)
35		funun 5275	. . 3 ⊢ (((Fun ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∧ Fun ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) ∧ (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) = ∅) → Fun (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))))
36	11, 22, 34, 35	syl21anc 1248	. 2 ⊢ (𝜑 → Fun (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))))
37		df-djud 7120	. . . . 5 ⊢ (𝑅 ⊔d 𝑆) = ((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
38	37	cnveqi 4817	. . . 4 ⊢ ◡(𝑅 ⊔d 𝑆) = ◡((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
39		cnvun 5049	. . . 4 ⊢ ◡((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆))) = (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
40	38, 39	eqtri 2210	. . 3 ⊢ ◡(𝑅 ⊔d 𝑆) = (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
41	40	funeqi 5252	. 2 ⊢ (Fun ◡(𝑅 ⊔d 𝑆) ↔ Fun (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))))
42	36, 41	sylibr 134	1 ⊢ (𝜑 → Fun ◡(𝑅 ⊔d 𝑆))

Syntax hints:   → wi 4   = wceq 1364   ∪ cun 3142   ∩ cin 3143   ⊆ wss 3144  ∅c0 3437  ^◡ccnv 4640  dom cdm 4641  ran crn 4642   ↾ cres 4643   ∘ ccom 4645  Fun wfun 5225  –1-1→wf1 5228   ⊔ cdju 7054  inlcinl 7062  inrcinr 7063   ⊔_d cdjud 7119

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-1st 6159  df-2nd 6160  df-1o 6435  df-dju 7055  df-inl 7064  df-inr 7065  df-djud 7120

This theorem is referenced by: (None)