Theorem djufun 6997

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 6954	. . . . 5 ⊢ (inl ↾ dom 𝐹):dom 𝐹–1-1→(dom 𝐹 ⊔ dom 𝐺)
3		df-f1 5136	. . . . . 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 5171	. . . 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 6955	. . . . 5 ⊢ (inr ↾ dom 𝐺):dom 𝐺–1-1→(dom 𝐹 ⊔ dom 𝐺)
10		df-f1 5136	. . . . . 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 5171	. . . 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 4816	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ dom ◡(inl ↾ dom 𝐹)
16		df-rn 4558	. . . . . . 7 ⊢ ran (inl ↾ dom 𝐹) = dom ◡(inl ↾ dom 𝐹)
17	15, 16	sseqtrri 3137	. . . . . 6 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18		dmcoss 4816	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ dom ◡(inr ↾ dom 𝐺)
19		df-rn 4558	. . . . . . 7 ⊢ ran (inr ↾ dom 𝐺) = dom ◡(inr ↾ dom 𝐺)
20	18, 19	sseqtrri 3137	. . . . . 6 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21		ss2in 3309	. . . . . 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 6956	. . . . . 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 3152	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ ∅)
26		ss0 3408	. . . 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 5175	. . 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 1216	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))))
30		df-djud 6996	. . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
31	30	funeqi 5152	. 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 1332   ∪ cun 3074   ∩ cin 3075   ⊆ wss 3076  ∅c0 3368  ^◡ccnv 4546  dom cdm 4547  ran crn 4548   ↾ cres 4549   ∘ ccom 4551  Fun wfun 5125  ⟶wf 5127  –1-1→wf1 5128   ⊔ cdju 6930  inlcinl 6938  inrcinr 6939   ⊔_d cdjud 6995

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-dju 6931  df-inl 6940  df-inr 6941  df-djud 6996

This theorem is referenced by: (None)