Theorem casefun 7144

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 7117	. . . . . 6 ⊢ inl:V–1-1-onto→({∅} × V)
3		f1of1 5499	. . . . . 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 5259	. . . . . 6 ⊢ (inl:V–1-1→({∅} × V) ↔ (inl:V⟶({∅} × V) ∧ Fun ◡inl))
6	5	simprbi 275	. . . . 5 ⊢ (inl:V–1-1→({∅} × V) → Fun ◡inl)
7	4, 6	mp1i 10	. . . 4 ⊢ (𝜑 → Fun ◡inl)
8		funco 5294	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
9	1, 7, 8	syl2anc 411	. . 3 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
10		casefun.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
11		djurf1o 7118	. . . . . 6 ⊢ inr:V–1-1-onto→({1o} × V)
12		f1of1 5499	. . . . . 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 5259	. . . . . 6 ⊢ (inr:V–1-1→({1o} × V) ↔ (inr:V⟶({1o} × V) ∧ Fun ◡inr))
15	14	simprbi 275	. . . . 5 ⊢ (inr:V–1-1→({1o} × V) → Fun ◡inr)
16	13, 15	mp1i 10	. . . 4 ⊢ (𝜑 → Fun ◡inr)
17		funco 5294	. . . 4 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
18	10, 16, 17	syl2anc 411	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
19		dmcoss 4931	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡inl) ⊆ dom ◡inl
20		df-rn 4670	. . . . . . 7 ⊢ ran inl = dom ◡inl
21	19, 20	sseqtrri 3214	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) ⊆ ran inl
22		dmcoss 4931	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) ⊆ dom ◡inr
23		df-rn 4670	. . . . . . 7 ⊢ ran inr = dom ◡inr
24	22, 23	sseqtrri 3214	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) ⊆ ran inr
25		ss2in 3387	. . . . . 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 426	. . . . 5 ⊢ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ (ran inl ∩ ran inr)
27		rnresv 5125	. . . . . . . . 9 ⊢ ran (inl ↾ V) = ran inl
28	27	eqcomi 2197	. . . . . . . 8 ⊢ ran inl = ran (inl ↾ V)
29		rnresv 5125	. . . . . . . . 9 ⊢ ran (inr ↾ V) = ran inr
30	29	eqcomi 2197	. . . . . . . 8 ⊢ ran inr = ran (inr ↾ V)
31	28, 30	ineq12i 3358	. . . . . . 7 ⊢ (ran inl ∩ ran inr) = (ran (inl ↾ V) ∩ ran (inr ↾ V))
32		djuinr 7122	. . . . . . 7 ⊢ (ran (inl ↾ V) ∩ ran (inr ↾ V)) = ∅
33	31, 32	eqtri 2214	. . . . . 6 ⊢ (ran inl ∩ ran inr) = ∅
34	33	a1i 9	. . . . 5 ⊢ (𝜑 → (ran inl ∩ ran inr) = ∅)
35	26, 34	sseqtrid 3229	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ ∅)
36		ss0 3487	. . . 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 5298	. . 3 ⊢ (((Fun (𝐹 ∘ ◡inl) ∧ Fun (𝐺 ∘ ◡inr)) ∧ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) = ∅) → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
39	9, 18, 37, 38	syl21anc 1248	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
40		df-case 7143	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
41	40	funeqi 5275	. 2 ⊢ (Fun case(𝐹, 𝐺) ↔ Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
42	39, 41	sylibr 134	1 ⊢ (𝜑 → Fun case(𝐹, 𝐺))

Syntax hints:   → wi 4   = wceq 1364  Vcvv 2760   ∪ cun 3151   ∩ cin 3152   ⊆ wss 3153  ∅c0 3446  {csn 3618   × cxp 4657  ^◡ccnv 4658  dom cdm 4659  ran crn 4660   ↾ cres 4661   ∘ ccom 4663  Fun wfun 5248  ⟶wf 5250  –1-1→wf1 5251  –1-1-onto→wf1o 5253  1_oc1o 6462  inlcinl 7104  inrcinr 7105  casecdjucase 7142

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-inl 7106  df-inr 7107  df-case 7143

This theorem is referenced by:  casef  7147