Theorem djufun 7163

Description: The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
djufun.f	⊢ (𝜑 → Fun 𝐹)
djufun.g	⊢ (𝜑 → Fun 𝐺)

Assertion

Ref	Expression
djufun	⊢ (𝜑 → Fun (𝐹 ⊔d 𝐺))

Proof of Theorem djufun

Step	Hyp	Ref	Expression
1		djufun.f	. . . 4 ⊢ (𝜑 → Fun 𝐹)
2		inlresf1 7120	. . . . 5 ⊢ (inl ↾ dom 𝐹):dom 𝐹–1-1→(dom 𝐹 ⊔ dom 𝐺)
3		df-f1 5259	. . . . . 6 ⊢ ((inl ↾ dom 𝐹):dom 𝐹–1-1→(dom 𝐹 ⊔ dom 𝐺) ↔ ((inl ↾ dom 𝐹):dom 𝐹⟶(dom 𝐹 ⊔ dom 𝐺) ∧ Fun ◡(inl ↾ dom 𝐹)))
4	3	simprbi 275	. . . . 5 ⊢ ((inl ↾ dom 𝐹):dom 𝐹–1-1→(dom 𝐹 ⊔ dom 𝐺) → Fun ◡(inl ↾ dom 𝐹))
5	2, 4	mp1i 10	. . . 4 ⊢ (𝜑 → Fun ◡(inl ↾ dom 𝐹))
6		funco 5294	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun ◡(inl ↾ dom 𝐹)) → Fun (𝐹 ∘ ◡(inl ↾ dom 𝐹)))
7	1, 5, 6	syl2anc 411	. . 3 ⊢ (𝜑 → Fun (𝐹 ∘ ◡(inl ↾ dom 𝐹)))
8		djufun.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
9		inrresf1 7121	. . . . 5 ⊢ (inr ↾ dom 𝐺):dom 𝐺–1-1→(dom 𝐹 ⊔ dom 𝐺)
10		df-f1 5259	. . . . . 6 ⊢ ((inr ↾ dom 𝐺):dom 𝐺–1-1→(dom 𝐹 ⊔ dom 𝐺) ↔ ((inr ↾ dom 𝐺):dom 𝐺⟶(dom 𝐹 ⊔ dom 𝐺) ∧ Fun ◡(inr ↾ dom 𝐺)))
11	10	simprbi 275	. . . . 5 ⊢ ((inr ↾ dom 𝐺):dom 𝐺–1-1→(dom 𝐹 ⊔ dom 𝐺) → Fun ◡(inr ↾ dom 𝐺))
12	9, 11	mp1i 10	. . . 4 ⊢ (𝜑 → Fun ◡(inr ↾ dom 𝐺))
13		funco 5294	. . . 4 ⊢ ((Fun 𝐺 ∧ Fun ◡(inr ↾ dom 𝐺)) → Fun (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
14	8, 12, 13	syl2anc 411	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
15		dmcoss 4931	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ dom ◡(inl ↾ dom 𝐹)
16		df-rn 4670	. . . . . . 7 ⊢ ran (inl ↾ dom 𝐹) = dom ◡(inl ↾ dom 𝐹)
17	15, 16	sseqtrri 3214	. . . . . 6 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18		dmcoss 4931	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ dom ◡(inr ↾ dom 𝐺)
19		df-rn 4670	. . . . . . 7 ⊢ ran (inr ↾ dom 𝐺) = dom ◡(inr ↾ dom 𝐺)
20	18, 19	sseqtrri 3214	. . . . . 6 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21		ss2in 3387	. . . . . 6 ⊢ ((dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹) ∧ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)) → (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)))
22	17, 20, 21	mp2an 426	. . . . 5 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺))
23		djuinr 7122	. . . . . 6 ⊢ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅
24	23	a1i 9	. . . . 5 ⊢ (𝜑 → (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅)
25	22, 24	sseqtrid 3229	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ ∅)
26		ss0 3487	. . . 4 ⊢ ((dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ ∅ → (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = ∅)
27	25, 26	syl 14	. . 3 ⊢ (𝜑 → (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = ∅)
28		funun 5298	. . 3 ⊢ (((Fun (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∧ Fun (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ∧ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) = ∅) → Fun ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))))
29	7, 14, 27, 28	syl21anc 1248	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))))
30		df-djud 7162	. . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
31	30	funeqi 5275	. 2 ⊢ (Fun (𝐹 ⊔d 𝐺) ↔ Fun ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))))
32	29, 31	sylibr 134	1 ⊢ (𝜑 → Fun (𝐹 ⊔d 𝐺))

Syntax hints:   → wi 4   = wceq 1364   ∪ cun 3151   ∩ cin 3152   ⊆ wss 3153  ∅c0 3446  ^◡ccnv 4658  dom cdm 4659  ran crn 4660   ↾ cres 4661   ∘ ccom 4663  Fun wfun 5248  ⟶wf 5250  –1-1→wf1 5251   ⊔ cdju 7096  inlcinl 7104  inrcinr 7105   ⊔_d cdjud 7161

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-dju 7097  df-inl 7106  df-inr 7107  df-djud 7162

This theorem is referenced by: (None)