Theorem djuinj 7365

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 7320	. . . . . . 7 ⊢ (inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ 𝐴)
2		f1fun 5554	. . . . . . 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 5396	. . . . . 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 5373	. . . . 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 4921	. . . . 5 ⊢ ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) = (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅)
10	9	funeqi 5354	. . . 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 7321	. . . . . . 7 ⊢ (inr ↾ dom 𝑆):dom 𝑆–1-1→(𝐴 ⊔ dom 𝑆)
13		f1fun 5554	. . . . . . 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 5396	. . . . . 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 5373	. . . . 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 4921	. . . . 5 ⊢ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) = (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆)
21	20	funeqi 5354	. . . 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 4742	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡(inl ↾ dom 𝑅)) = dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅))
24		rncoss 5009	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅
25	23, 24	eqsstrri 3261	. . . . . 6 ⊢ dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅
26		df-rn 4742	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡(inr ↾ dom 𝑆)) = dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))
27		rncoss 5009	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆
28	26, 27	eqsstrri 3261	. . . . . 6 ⊢ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆
29		ss2in 3437	. . . . . 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 3278	. . . 4 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) ⊆ ∅)
33		ss0 3537	. . . 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 5378	. . 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 1273	. 2 ⊢ (𝜑 → Fun (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))))
37		df-djud 7362	. . . . 5 ⊢ (𝑅 ⊔d 𝑆) = ((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
38	37	cnveqi 4911	. . . 4 ⊢ ◡(𝑅 ⊔d 𝑆) = ◡((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
39		cnvun 5149	. . . 4 ⊢ ◡((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆))) = (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
40	38, 39	eqtri 2252	. . 3 ⊢ ◡(𝑅 ⊔d 𝑆) = (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
41	40	funeqi 5354	. 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 1398   ∪ cun 3199   ∩ cin 3200   ⊆ wss 3201  ∅c0 3496  ^◡ccnv 4730  dom cdm 4731  ran crn 4732   ↾ cres 4733   ∘ ccom 4735  Fun wfun 5327  –1-1→wf1 5330   ⊔ cdju 7296  inlcinl 7304  inrcinr 7305   ⊔_d cdjud 7361

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-dju 7297  df-inl 7306  df-inr 7307  df-djud 7362

This theorem is referenced by: (None)