Theorem djufun 6989

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 6946	. . . . 5 ⊢ (inl ↾ dom 𝐹):dom 𝐹–1-1→(dom 𝐹 ⊔ dom 𝐺)
3		df-f1 5128	. . . . . 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 5163	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun ◡(inl ↾ dom 𝐹)) → Fun (𝐹 ∘ ◡(inl ↾ dom 𝐹)))
7	1, 5, 6	syl2anc 408	. . 3 ⊢ (𝜑 → Fun (𝐹 ∘ ◡(inl ↾ dom 𝐹)))
8		djufun.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
9		inrresf1 6947	. . . . 5 ⊢ (inr ↾ dom 𝐺):dom 𝐺–1-1→(dom 𝐹 ⊔ dom 𝐺)
10		df-f1 5128	. . . . . 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 5163	. . . 4 ⊢ ((Fun 𝐺 ∧ Fun ◡(inr ↾ dom 𝐺)) → Fun (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
14	8, 12, 13	syl2anc 408	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
15		dmcoss 4808	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ dom ◡(inl ↾ dom 𝐹)
16		df-rn 4550	. . . . . . 7 ⊢ ran (inl ↾ dom 𝐹) = dom ◡(inl ↾ dom 𝐹)
17	15, 16	sseqtrri 3132	. . . . . 6 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18		dmcoss 4808	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ dom ◡(inr ↾ dom 𝐺)
19		df-rn 4550	. . . . . . 7 ⊢ ran (inr ↾ dom 𝐺) = dom ◡(inr ↾ dom 𝐺)
20	18, 19	sseqtrri 3132	. . . . . 6 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21		ss2in 3304	. . . . . 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 422	. . . . 5 ⊢ (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺))
23		djuinr 6948	. . . . . 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 3147	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ ∅)
26		ss0 3403	. . . 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 5167	. . 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 1215	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))))
30		df-djud 6988	. . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
31	30	funeqi 5144	. 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 1331   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  ^◡ccnv 4538  dom cdm 4539  ran crn 4540   ↾ cres 4541   ∘ ccom 4543  Fun wfun 5117  ⟶wf 5119  –1-1→wf1 5120   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931   ⊔_d cdjud 6987

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933  df-djud 6988

This theorem is referenced by: (None)