Theorem djufun 7069

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 7026	. . . . 5 ⊢ (inl ↾ dom 𝐹):dom 𝐹–1-1→(dom 𝐹 ⊔ dom 𝐺)
3		df-f1 5193	. . . . . 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 273	. . . . 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 5228	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun ◡(inl ↾ dom 𝐹)) → Fun (𝐹 ∘ ◡(inl ↾ dom 𝐹)))
7	1, 5, 6	syl2anc 409	. . 3 ⊢ (𝜑 → Fun (𝐹 ∘ ◡(inl ↾ dom 𝐹)))
8		djufun.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
9		inrresf1 7027	. . . . 5 ⊢ (inr ↾ dom 𝐺):dom 𝐺–1-1→(dom 𝐹 ⊔ dom 𝐺)
10		df-f1 5193	. . . . . 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 273	. . . . 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 5228	. . . 4 ⊢ ((Fun 𝐺 ∧ Fun ◡(inr ↾ dom 𝐺)) → Fun (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
14	8, 12, 13	syl2anc 409	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
15		dmcoss 4873	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ dom ◡(inl ↾ dom 𝐹)
16		df-rn 4615	. . . . . . 7 ⊢ ran (inl ↾ dom 𝐹) = dom ◡(inl ↾ dom 𝐹)
17	15, 16	sseqtrri 3177	. . . . . 6 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18		dmcoss 4873	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ dom ◡(inr ↾ dom 𝐺)
19		df-rn 4615	. . . . . . 7 ⊢ ran (inr ↾ dom 𝐺) = dom ◡(inr ↾ dom 𝐺)
20	18, 19	sseqtrri 3177	. . . . . 6 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21		ss2in 3350	. . . . . 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 423	. . . . 5 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺))
23		djuinr 7028	. . . . . 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 3192	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ ∅)
26		ss0 3449	. . . 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 5232	. . 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 1227	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))))
30		df-djud 7068	. . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
31	30	funeqi 5209	. 2 ⊢ (Fun (𝐹 ⊔d 𝐺) ↔ Fun ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))))
32	29, 31	sylibr 133	1 ⊢ (𝜑 → Fun (𝐹 ⊔d 𝐺))

Syntax hints:   → wi 4   = wceq 1343   ∪ cun 3114   ∩ cin 3115   ⊆ wss 3116  ∅c0 3409  ^◡ccnv 4603  dom cdm 4604  ran crn 4605   ↾ cres 4606   ∘ ccom 4608  Fun wfun 5182  ⟶wf 5184  –1-1→wf1 5185   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011   ⊔_d cdjud 7067

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013  df-djud 7068

This theorem is referenced by: (None)