Theorem casefun 7389

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 7362	. . . . . 6 ⊢ inl:V–1-1-onto→({∅} × V)
3		f1of1 5618	. . . . . 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 5362	. . . . . 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 5397	. . . 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 7363	. . . . . 6 ⊢ inr:V–1-1-onto→({1o} × V)
12		f1of1 5618	. . . . . 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 5362	. . . . . 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 5397	. . . 4 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
18	10, 16, 17	syl2anc 411	. . 3 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
19		dmcoss 5032	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡inl) ⊆ dom ◡inl
20		df-rn 4765	. . . . . . 7 ⊢ ran inl = dom ◡inl
21	19, 20	sseqtrri 3277	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) ⊆ ran inl
22		dmcoss 5032	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) ⊆ dom ◡inr
23		df-rn 4765	. . . . . . 7 ⊢ ran inr = dom ◡inr
24	22, 23	sseqtrri 3277	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) ⊆ ran inr
25		ss2in 3453	. . . . . 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 5227	. . . . . . . . 9 ⊢ ran (inl ↾ V) = ran inl
28	27	eqcomi 2238	. . . . . . . 8 ⊢ ran inl = ran (inl ↾ V)
29		rnresv 5227	. . . . . . . . 9 ⊢ ran (inr ↾ V) = ran inr
30	29	eqcomi 2238	. . . . . . . 8 ⊢ ran inr = ran (inr ↾ V)
31	28, 30	ineq12i 3424	. . . . . . 7 ⊢ (ran inl ∩ ran inr) = (ran (inl ↾ V) ∩ ran (inr ↾ V))
32		djuinr 7367	. . . . . . 7 ⊢ (ran (inl ↾ V) ∩ ran (inr ↾ V)) = ∅
33	31, 32	eqtri 2255	. . . . . 6 ⊢ (ran inl ∩ ran inr) = ∅
34	33	a1i 9	. . . . 5 ⊢ (𝜑 → (ran inl ∩ ran inr) = ∅)
35	26, 34	sseqtrid 3292	. . . 4 ⊢ (𝜑 → (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) ⊆ ∅)
36		ss0 3553	. . . 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 5402	. . 3 ⊢ (((Fun (𝐹 ∘ ◡inl) ∧ Fun (𝐺 ∘ ◡inr)) ∧ (dom (𝐹 ∘ ◡inl) ∩ dom (𝐺 ∘ ◡inr)) = ∅) → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
39	9, 18, 37, 38	syl21anc 1273	. 2 ⊢ (𝜑 → Fun ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr)))
40		df-case 7388	. . 3 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
41	40	funeqi 5378	. 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 1398  Vcvv 2815   ∪ cun 3212   ∩ cin 3213   ⊆ wss 3214  ∅c0 3512  {csn 3694   × cxp 4752  ^◡ccnv 4753  dom cdm 4754  ran crn 4755   ↾ cres 4756   ∘ ccom 4758  Fun wfun 5351  ⟶wf 5353  –1-1→wf1 5354  –1-1-onto→wf1o 5356  1_oc1o 6653  inlcinl 7349  inrcinr 7350  casecdjucase 7387

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-inl 7351  df-inr 7352  df-case 7388

This theorem is referenced by:  casef  7392