Theorem casefun 6970

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 6943	. . . . . 6 ⊢ inl:V–1-1-onto→({∅} × V)
3		f1of1 5366	. . . . . 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 5128	. . . . . 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 5163	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
9	1, 7, 8	syl2anc 408	. . 3 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
10		casefun.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
11		djurf1o 6944	. . . . . 6 ⊢ inr:V–1-1-onto→({1o} × V)
12		f1of1 5366	. . . . . 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 5128	. . . . . 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 5163	. . . 4 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
18	10, 16, 17	syl2anc 408	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
19		dmcoss 4808	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡inl) ⊆ dom ◡inl
20		df-rn 4550	. . . . . . 7 ⊢ ran inl = dom ◡inl
21	19, 20	sseqtrri 3132	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) ⊆ ran inl
22		dmcoss 4808	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) ⊆ dom ◡inr
23		df-rn 4550	. . . . . . 7 ⊢ ran inr = dom ◡inr
24	22, 23	sseqtrri 3132	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) ⊆ ran inr
25		ss2in 3304	. . . . . 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 422	. . . . 5 ⊢ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ (ran inl ∩ ran inr)
27		rnresv 4998	. . . . . . . . 9 ⊢ ran (inl ↾ V) = ran inl
28	27	eqcomi 2143	. . . . . . . 8 ⊢ ran inl = ran (inl ↾ V)
29		rnresv 4998	. . . . . . . . 9 ⊢ ran (inr ↾ V) = ran inr
30	29	eqcomi 2143	. . . . . . . 8 ⊢ ran inr = ran (inr ↾ V)
31	28, 30	ineq12i 3275	. . . . . . 7 ⊢ (ran inl ∩ ran inr) = (ran (inl ↾ V) ∩ ran (inr ↾ V))
32		djuinr 6948	. . . . . . 7 ⊢ (ran (inl ↾ V) ∩ ran (inr ↾ V)) = ∅
33	31, 32	eqtri 2160	. . . . . 6 ⊢ (ran inl ∩ ran inr) = ∅
34	33	a1i 9	. . . . 5 ⊢ (𝜑 → (ran inl ∩ ran inr) = ∅)
35	26, 34	sseqtrid 3147	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ ∅)
36		ss0 3403	. . . 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 5167	. . 3 ⊢ (((Fun (𝐹 ∘ ◡inl) ∧ Fun (𝐺 ∘ ◡inr)) ∧ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) = ∅) → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
39	9, 18, 37, 38	syl21anc 1215	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
40		df-case 6969	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
41	40	funeqi 5144	. 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 1331  Vcvv 2686   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  {csn 3527   × cxp 4537  ^◡ccnv 4538  dom cdm 4539  ran crn 4540   ↾ cres 4541   ∘ ccom 4543  Fun wfun 5117  ⟶wf 5119  –1-1→wf1 5120  –1-1-onto→wf1o 5122  1_oc1o 6306  inlcinl 6930  inrcinr 6931  casecdjucase 6968

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-inl 6932  df-inr 6933  df-case 6969

This theorem is referenced by:  casef  6973