Theorem djufun 7232

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 7189	. . . . 5 ⊢ (inl ↾ dom 𝐹):dom 𝐹–1-1→(dom 𝐹 ⊔ dom 𝐺)
3		df-f1 5295	. . . . . 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 5330	. . . 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 7190	. . . . 5 ⊢ (inr ↾ dom 𝐺):dom 𝐺–1-1→(dom 𝐹 ⊔ dom 𝐺)
10		df-f1 5295	. . . . . 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 5330	. . . 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 4967	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ dom ◡(inl ↾ dom 𝐹)
16		df-rn 4704	. . . . . . 7 ⊢ ran (inl ↾ dom 𝐹) = dom ◡(inl ↾ dom 𝐹)
17	15, 16	sseqtrri 3236	. . . . . 6 ⊢ dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18		dmcoss 4967	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ dom ◡(inr ↾ dom 𝐺)
19		df-rn 4704	. . . . . . 7 ⊢ ran (inr ↾ dom 𝐺) = dom ◡(inr ↾ dom 𝐺)
20	18, 19	sseqtrri 3236	. . . . . 6 ⊢ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21		ss2in 3409	. . . . . 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 7191	. . . . . 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 3251	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∩ dom (𝐺 ∘ ◡(inr ↾ dom 𝐺))) ⊆ ∅)
26		ss0 3509	. . . 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 5334	. . 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 1249	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺))))
30		df-djud 7231	. . 3 ⊢ (𝐹 ⊔d 𝐺) = ((𝐹 ∘ ◡(inl ↾ dom 𝐹)) ∪ (𝐺 ∘ ◡(inr ↾ dom 𝐺)))
31	30	funeqi 5311	. 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 1373   ∪ cun 3172   ∩ cin 3173   ⊆ wss 3174  ∅c0 3468  ^◡ccnv 4692  dom cdm 4693  ran crn 4694   ↾ cres 4695   ∘ ccom 4697  Fun wfun 5284  ⟶wf 5286  –1-1→wf1 5287   ⊔ cdju 7165  inlcinl 7173  inrcinr 7174   ⊔_d cdjud 7230

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-1o 6525  df-dju 7166  df-inl 7175  df-inr 7176  df-djud 7231

This theorem is referenced by: (None)