Theorem djuinj 7051

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 7006	. . . . . . 7 ⊢ (inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ 𝐴)
2		f1fun 5379	. . . . . . 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 5230	. . . . . 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 5211	. . . . 5 ⊢ ((Fun ◡◡(inl ↾ dom 𝑅) ∧ Fun ◡𝑅) → Fun (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅))
8	5, 6, 7	sylancr 411	. . . 4 ⊢ (𝜑 → Fun (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅))
9		cnvco 4772	. . . . 5 ⊢ ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) = (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅)
10	9	funeqi 5192	. . . 4 ⊢ (Fun ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ↔ Fun (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅))
11	8, 10	sylibr 133	. . 3 ⊢ (𝜑 → Fun ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)))
12		inrresf1 7007	. . . . . . 7 ⊢ (inr ↾ dom 𝑆):dom 𝑆–1-1→(𝐴 ⊔ dom 𝑆)
13		f1fun 5379	. . . . . . 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 5230	. . . . . 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 5211	. . . . 5 ⊢ ((Fun ◡◡(inr ↾ dom 𝑆) ∧ Fun ◡𝑆) → Fun (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆))
19	16, 17, 18	sylancr 411	. . . 4 ⊢ (𝜑 → Fun (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆))
20		cnvco 4772	. . . . 5 ⊢ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) = (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆)
21	20	funeqi 5192	. . . 4 ⊢ (Fun ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) ↔ Fun (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆))
22	19, 21	sylibr 133	. . 3 ⊢ (𝜑 → Fun ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
23		df-rn 4598	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡(inl ↾ dom 𝑅)) = dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅))
24		rncoss 4857	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅
25	23, 24	eqsstrri 3161	. . . . . 6 ⊢ dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅
26		df-rn 4598	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡(inr ↾ dom 𝑆)) = dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))
27		rncoss 4857	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆
28	26, 27	eqsstrri 3161	. . . . . 6 ⊢ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆
29		ss2in 3335	. . . . . 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 423	. . . . 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 3178	. . . 4 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) ⊆ ∅)
33		ss0 3434	. . . 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 5215	. . 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 1219	. 2 ⊢ (𝜑 → Fun (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))))
37		df-djud 7048	. . . . 5 ⊢ (𝑅 ⊔d 𝑆) = ((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
38	37	cnveqi 4762	. . . 4 ⊢ ◡(𝑅 ⊔d 𝑆) = ◡((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
39		cnvun 4992	. . . 4 ⊢ ◡((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆))) = (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
40	38, 39	eqtri 2178	. . 3 ⊢ ◡(𝑅 ⊔d 𝑆) = (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
41	40	funeqi 5192	. 2 ⊢ (Fun ◡(𝑅 ⊔d 𝑆) ↔ Fun (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))))
42	36, 41	sylibr 133	1 ⊢ (𝜑 → Fun ◡(𝑅 ⊔d 𝑆))

Syntax hints:   → wi 4   = wceq 1335   ∪ cun 3100   ∩ cin 3101   ⊆ wss 3102  ∅c0 3394  ^◡ccnv 4586  dom cdm 4587  ran crn 4588   ↾ cres 4589   ∘ ccom 4591  Fun wfun 5165  –1-1→wf1 5168   ⊔ cdju 6982  inlcinl 6990  inrcinr 6991   ⊔_d cdjud 7047

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-suc 4332  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-1st 6089  df-2nd 6090  df-1o 6364  df-dju 6983  df-inl 6992  df-inr 6993  df-djud 7048

This theorem is referenced by: (None)