Theorem casefun 7024

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 6997	. . . . . 6 ⊢ inl:V–1-1-onto→({∅} × V)
3		f1of1 5412	. . . . . 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 5174	. . . . . 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 5209	. . . 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 6998	. . . . . 6 ⊢ inr:V–1-1-onto→({1o} × V)
12		f1of1 5412	. . . . . 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 5174	. . . . . 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 5209	. . . 4 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
18	10, 16, 17	syl2anc 409	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
19		dmcoss 4854	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡inl) ⊆ dom ◡inl
20		df-rn 4596	. . . . . . 7 ⊢ ran inl = dom ◡inl
21	19, 20	sseqtrri 3163	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) ⊆ ran inl
22		dmcoss 4854	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) ⊆ dom ◡inr
23		df-rn 4596	. . . . . . 7 ⊢ ran inr = dom ◡inr
24	22, 23	sseqtrri 3163	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) ⊆ ran inr
25		ss2in 3335	. . . . . 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 423	. . . . 5 ⊢ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ (ran inl ∩ ran inr)
27		rnresv 5044	. . . . . . . . 9 ⊢ ran (inl ↾ V) = ran inl
28	27	eqcomi 2161	. . . . . . . 8 ⊢ ran inl = ran (inl ↾ V)
29		rnresv 5044	. . . . . . . . 9 ⊢ ran (inr ↾ V) = ran inr
30	29	eqcomi 2161	. . . . . . . 8 ⊢ ran inr = ran (inr ↾ V)
31	28, 30	ineq12i 3306	. . . . . . 7 ⊢ (ran inl ∩ ran inr) = (ran (inl ↾ V) ∩ ran (inr ↾ V))
32		djuinr 7002	. . . . . . 7 ⊢ (ran (inl ↾ V) ∩ ran (inr ↾ V)) = ∅
33	31, 32	eqtri 2178	. . . . . 6 ⊢ (ran inl ∩ ran inr) = ∅
34	33	a1i 9	. . . . 5 ⊢ (𝜑 → (ran inl ∩ ran inr) = ∅)
35	26, 34	sseqtrid 3178	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ ∅)
36		ss0 3434	. . . 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 5213	. . 3 ⊢ (((Fun (𝐹 ∘ ◡inl) ∧ Fun (𝐺 ∘ ◡inr)) ∧ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) = ∅) → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
39	9, 18, 37, 38	syl21anc 1219	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
40		df-case 7023	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
41	40	funeqi 5190	. 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 1335  Vcvv 2712   ∪ cun 3100   ∩ cin 3101   ⊆ wss 3102  ∅c0 3394  {csn 3560   × cxp 4583  ^◡ccnv 4584  dom cdm 4585  ran crn 4586   ↾ cres 4587   ∘ ccom 4589  Fun wfun 5163  ⟶wf 5165  –1-1→wf1 5166  –1-1-onto→wf1o 5168  1_oc1o 6353  inlcinl 6984  inrcinr 6985  casecdjucase 7022

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-1st 6085  df-2nd 6086  df-1o 6360  df-inl 6986  df-inr 6987  df-case 7023

This theorem is referenced by:  casef  7027