Theorem casefun 7062

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

Hypotheses

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

Assertion

Ref	Expression
casefun	⊢ (𝜑 → Fun case(𝐹, 𝐺))

Proof of Theorem casefun

Step	Hyp	Ref	Expression
1		casefun.f	. . . 4 ⊢ (𝜑 → Fun 𝐹)
2		djulf1o 7035	. . . . . 6 ⊢ inl:V–1-1-onto→({∅} × V)
3		f1of1 5441	. . . . . 6 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
4	2, 3	ax-mp 5	. . . . 5 ⊢ inl:V–1-1→({∅} × V)
5		df-f1 5203	. . . . . 6 ⊢ (inl:V–1-1→({∅} × V) ↔ (inl:V⟶({∅} × V) ∧ Fun ◡inl))
6	5	simprbi 273	. . . . 5 ⊢ (inl:V–1-1→({∅} × V) → Fun ◡inl)
7	4, 6	mp1i 10	. . . 4 ⊢ (𝜑 → Fun ◡inl)
8		funco 5238	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
9	1, 7, 8	syl2anc 409	. . 3 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
10		casefun.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
11		djurf1o 7036	. . . . . 6 ⊢ inr:V–1-1-onto→({1o} × V)
12		f1of1 5441	. . . . . 6 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V–1-1→({1o} × V))
13	11, 12	ax-mp 5	. . . . 5 ⊢ inr:V–1-1→({1o} × V)
14		df-f1 5203	. . . . . 6 ⊢ (inr:V–1-1→({1o} × V) ↔ (inr:V⟶({1o} × V) ∧ Fun ◡inr))
15	14	simprbi 273	. . . . 5 ⊢ (inr:V–1-1→({1o} × V) → Fun ◡inr)
16	13, 15	mp1i 10	. . . 4 ⊢ (𝜑 → Fun ◡inr)
17		funco 5238	. . . 4 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
18	10, 16, 17	syl2anc 409	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
19		dmcoss 4880	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡inl) ⊆ dom ◡inl
20		df-rn 4622	. . . . . . 7 ⊢ ran inl = dom ◡inl
21	19, 20	sseqtrri 3182	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) ⊆ ran inl
22		dmcoss 4880	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) ⊆ dom ◡inr
23		df-rn 4622	. . . . . . 7 ⊢ ran inr = dom ◡inr
24	22, 23	sseqtrri 3182	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) ⊆ ran inr
25		ss2in 3355	. . . . . 6 ⊢ ((dom (𝐹 ∘ ◡inl) ⊆ ran inl ∧ dom (𝐺 ∘ ◡inr) ⊆ ran inr) → (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ (ran inl ∩ ran inr))
26	21, 24, 25	mp2an 424	. . . . 5 ⊢ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ (ran inl ∩ ran inr)
27		rnresv 5070	. . . . . . . . 9 ⊢ ran (inl ↾ V) = ran inl
28	27	eqcomi 2174	. . . . . . . 8 ⊢ ran inl = ran (inl ↾ V)
29		rnresv 5070	. . . . . . . . 9 ⊢ ran (inr ↾ V) = ran inr
30	29	eqcomi 2174	. . . . . . . 8 ⊢ ran inr = ran (inr ↾ V)
31	28, 30	ineq12i 3326	. . . . . . 7 ⊢ (ran inl ∩ ran inr) = (ran (inl ↾ V) ∩ ran (inr ↾ V))
32		djuinr 7040	. . . . . . 7 ⊢ (ran (inl ↾ V) ∩ ran (inr ↾ V)) = ∅
33	31, 32	eqtri 2191	. . . . . 6 ⊢ (ran inl ∩ ran inr) = ∅
34	33	a1i 9	. . . . 5 ⊢ (𝜑 → (ran inl ∩ ran inr) = ∅)
35	26, 34	sseqtrid 3197	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ ∅)
36		ss0 3455	. . . 4 ⊢ ((dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ ∅ → (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) = ∅)
37	35, 36	syl 14	. . 3 ⊢ (𝜑 → (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) = ∅)
38		funun 5242	. . 3 ⊢ (((Fun (𝐹 ∘ ◡inl) ∧ Fun (𝐺 ∘ ◡inr)) ∧ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) = ∅) → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
39	9, 18, 37, 38	syl21anc 1232	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
40		df-case 7061	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
41	40	funeqi 5219	. 2 ⊢ (Fun case(𝐹, 𝐺) ↔ Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
42	39, 41	sylibr 133	1 ⊢ (𝜑 → Fun case(𝐹, 𝐺))

Syntax hints:   → wi 4   = wceq 1348  Vcvv 2730   ∪ cun 3119   ∩ cin 3120   ⊆ wss 3121  ∅c0 3414  {csn 3583   × cxp 4609  ^◡ccnv 4610  dom cdm 4611  ran crn 4612   ↾ cres 4613   ∘ ccom 4615  Fun wfun 5192  ⟶wf 5194  –1-1→wf1 5195  –1-1-onto→wf1o 5197  1_oc1o 6388  inlcinl 7022  inrcinr 7023  casecdjucase 7060

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-inl 7024  df-inr 7025  df-case 7061

This theorem is referenced by:  casef  7065