Theorem djuinj 7436

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 7391	. . . . . . 7 ⊢ (inl ↾ dom 𝑅):dom 𝑅–1-1→(dom 𝑅 ⊔ 𝐴)
2		f1fun 5596	. . . . . . 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 5435	. . . . . 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 5412	. . . . 5 ⊢ ((Fun ◡◡(inl ↾ dom 𝑅) ∧ Fun ◡𝑅) → Fun (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅))
8	5, 6, 7	sylancr 418	. . . 4 ⊢ (𝜑 → Fun (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅))
9		cnvco 4960	. . . . 5 ⊢ ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) = (◡◡(inl ↾ dom 𝑅) ∘ ◡𝑅)
10	9	funeqi 5393	. . . 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 7392	. . . . . . 7 ⊢ (inr ↾ dom 𝑆):dom 𝑆–1-1→(𝐴 ⊔ dom 𝑆)
13		f1fun 5596	. . . . . . 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 5435	. . . . . 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 5412	. . . . 5 ⊢ ((Fun ◡◡(inr ↾ dom 𝑆) ∧ Fun ◡𝑆) → Fun (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆))
19	16, 17, 18	sylancr 418	. . . 4 ⊢ (𝜑 → Fun (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆))
20		cnvco 4960	. . . . 5 ⊢ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) = (◡◡(inr ↾ dom 𝑆) ∘ ◡𝑆)
21	20	funeqi 5393	. . . 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 4780	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡(inl ↾ dom 𝑅)) = dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅))
24		rncoss 5048	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅
25	23, 24	eqsstrri 3281	. . . . . 6 ⊢ dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ⊆ ran 𝑅
26		df-rn 4780	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡(inr ↾ dom 𝑆)) = dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))
27		rncoss 5048	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆
28	26, 27	eqsstrri 3281	. . . . . 6 ⊢ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)) ⊆ ran 𝑆
29		ss2in 3459	. . . . . 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 430	. . . . 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 3298	. . . 4 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∩ dom ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))) ⊆ ∅)
33		ss0 3563	. . . 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 5417	. . 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 1277	. 2 ⊢ (𝜑 → Fun (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆))))
37		df-djud 7433	. . . . 5 ⊢ (𝑅 ⊔d 𝑆) = ((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
38	37	cnveqi 4950	. . . 4 ⊢ ◡(𝑅 ⊔d 𝑆) = ◡((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
39		cnvun 5188	. . . 4 ⊢ ◡((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆))) = (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
40	38, 39	eqtri 2259	. . 3 ⊢ ◡(𝑅 ⊔d 𝑆) = (◡(𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ ◡(𝑆 ∘ ◡(inr ↾ dom 𝑆)))
41	40	funeqi 5393	. 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 1402   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  ^◡ccnv 4768  dom cdm 4769  ran crn 4770   ↾ cres 4771   ∘ ccom 4773  Fun wfun 5366  –1-1→wf1 5369   ⊔ cdju 7367  inlcinl 7375  inrcinr 7376   ⊔_d cdjud 7432

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-dju 7368  df-inl 7377  df-inr 7378  df-djud 7433

This theorem is referenced by: (None)