Theorem caseinl 7150

Description: Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)

Hypotheses

Ref	Expression
caseinl.f	⊢ (𝜑 → 𝐹 Fn 𝐵)
caseinl.g	⊢ (𝜑 → Fun 𝐺)
caseinl.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)

Assertion

Ref	Expression
caseinl	⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹‘𝐴))

Proof of Theorem caseinl

Step	Hyp	Ref	Expression
1		df-case 7143	. . . 4 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	fveq1i 5555	. . 3 ⊢ (case(𝐹, 𝐺)‘(inl‘𝐴)) = (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inl‘𝐴))
3		caseinl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐵)
4		fnfun 5351	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
6		djulf1o 7117	. . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V)
7		f1ocnv 5513	. . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → ◡inl:({∅} × V)–1-1-onto→V)
8	6, 7	ax-mp 5	. . . . . . 7 ⊢ ◡inl:({∅} × V)–1-1-onto→V
9		f1ofun 5502	. . . . . . 7 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun ◡inl
11		funco 5294	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
12	5, 10, 11	sylancl 413	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
13		dmco 5174	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
14		df-inl 7106	. . . . . . . . . . 11 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
15	14	funmpt2 5293	. . . . . . . . . 10 ⊢ Fun inl
16		funrel 5271	. . . . . . . . . 10 ⊢ (Fun inl → Rel inl)
17	15, 16	ax-mp 5	. . . . . . . . 9 ⊢ Rel inl
18		dfrel2 5116	. . . . . . . . 9 ⊢ (Rel inl ↔ ◡◡inl = inl)
19	17, 18	mpbi 145	. . . . . . . 8 ⊢ ◡◡inl = inl
20	19	a1i 9	. . . . . . 7 ⊢ (𝜑 → ◡◡inl = inl)
21		fndm 5353	. . . . . . . 8 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
22	3, 21	syl 14	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐵)
23	20, 22	imaeq12d 5006	. . . . . 6 ⊢ (𝜑 → (◡◡inl “ dom 𝐹) = (inl “ 𝐵))
24	13, 23	eqtrid 2238	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∘ ◡inl) = (inl “ 𝐵))
25		df-fn 5257	. . . . 5 ⊢ ((𝐹 ∘ ◡inl) Fn (inl “ 𝐵) ↔ (Fun (𝐹 ∘ ◡inl) ∧ dom (𝐹 ∘ ◡inl) = (inl “ 𝐵)))
26	12, 24, 25	sylanbrc 417	. . . 4 ⊢ (𝜑 → (𝐹 ∘ ◡inl) Fn (inl “ 𝐵))
27		caseinl.g	. . . . . 6 ⊢ (𝜑 → Fun 𝐺)
28		djurf1o 7118	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
29		f1ocnv 5513	. . . . . . . 8 ⊢ (inr:V–1-1-onto→({1o} × V) → ◡inr:({1o} × V)–1-1-onto→V)
30	28, 29	ax-mp 5	. . . . . . 7 ⊢ ◡inr:({1o} × V)–1-1-onto→V
31		f1ofun 5502	. . . . . . 7 ⊢ (◡inr:({1o} × V)–1-1-onto→V → Fun ◡inr)
32	30, 31	ax-mp 5	. . . . . 6 ⊢ Fun ◡inr
33		funco 5294	. . . . . 6 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
34	27, 32, 33	sylancl 413	. . . . 5 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
35		dmco 5174	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
36		imacnvcnv 5130	. . . . . . 7 ⊢ (◡◡inr “ dom 𝐺) = (inr “ dom 𝐺)
37	35, 36	eqtri 2214	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺)
38	37	a1i 9	. . . . 5 ⊢ (𝜑 → dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺))
39		df-fn 5257	. . . . 5 ⊢ ((𝐺 ∘ ◡inr) Fn (inr “ dom 𝐺) ↔ (Fun (𝐺 ∘ ◡inr) ∧ dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺)))
40	34, 38, 39	sylanbrc 417	. . . 4 ⊢ (𝜑 → (𝐺 ∘ ◡inr) Fn (inr “ dom 𝐺))
41		djuin 7123	. . . . 5 ⊢ ((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅
42	41	a1i 9	. . . 4 ⊢ (𝜑 → ((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅)
43		caseinl.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
44	43	elexd 2773	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
45		f1odm 5504	. . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → dom inl = V)
46	6, 45	ax-mp 5	. . . . . . 7 ⊢ dom inl = V
47	44, 46	eleqtrrdi 2287	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom inl)
48	47, 15	jctil 312	. . . . 5 ⊢ (𝜑 → (Fun inl ∧ 𝐴 ∈ dom inl))
49		funfvima 5790	. . . . 5 ⊢ ((Fun inl ∧ 𝐴 ∈ dom inl) → (𝐴 ∈ 𝐵 → (inl‘𝐴) ∈ (inl “ 𝐵)))
50	48, 43, 49	sylc 62	. . . 4 ⊢ (𝜑 → (inl‘𝐴) ∈ (inl “ 𝐵))
51		fvun1 5623	. . . 4 ⊢ (((𝐹 ∘ ◡inl) Fn (inl “ 𝐵) ∧ (𝐺 ∘ ◡inr) Fn (inr “ dom 𝐺) ∧ (((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅ ∧ (inl‘𝐴) ∈ (inl “ 𝐵))) → (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inl‘𝐴)) = ((𝐹 ∘ ◡inl)‘(inl‘𝐴)))
52	26, 40, 42, 50, 51	syl112anc 1253	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inl‘𝐴)) = ((𝐹 ∘ ◡inl)‘(inl‘𝐴)))
53	2, 52	eqtrid 2238	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = ((𝐹 ∘ ◡inl)‘(inl‘𝐴)))
54		f1ofn 5501	. . . 4 ⊢ (◡inl:({∅} × V)–1-1-onto→V → ◡inl Fn ({∅} × V))
55	8, 54	ax-mp 5	. . 3 ⊢ ◡inl Fn ({∅} × V)
56		f1of 5500	. . . . . 6 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
57	6, 56	ax-mp 5	. . . . 5 ⊢ inl:V⟶({∅} × V)
58	57	a1i 9	. . . 4 ⊢ (𝜑 → inl:V⟶({∅} × V))
59	58, 44	ffvelcdmd 5694	. . 3 ⊢ (𝜑 → (inl‘𝐴) ∈ ({∅} × V))
60		fvco2 5626	. . 3 ⊢ ((◡inl Fn ({∅} × V) ∧ (inl‘𝐴) ∈ ({∅} × V)) → ((𝐹 ∘ ◡inl)‘(inl‘𝐴)) = (𝐹‘(◡inl‘(inl‘𝐴))))
61	55, 59, 60	sylancr 414	. 2 ⊢ (𝜑 → ((𝐹 ∘ ◡inl)‘(inl‘𝐴)) = (𝐹‘(◡inl‘(inl‘𝐴))))
62		f1ocnvfv1 5820	. . . 4 ⊢ ((inl:V–1-1-onto→({∅} × V) ∧ 𝐴 ∈ V) → (◡inl‘(inl‘𝐴)) = 𝐴)
63	6, 44, 62	sylancr 414	. . 3 ⊢ (𝜑 → (◡inl‘(inl‘𝐴)) = 𝐴)
64	63	fveq2d 5558	. 2 ⊢ (𝜑 → (𝐹‘(◡inl‘(inl‘𝐴))) = (𝐹‘𝐴))
65	53, 61, 64	3eqtrd 2230	1 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∪ cun 3151   ∩ cin 3152  ∅c0 3446  {csn 3618  ⟨cop 3621   × cxp 4657  ^◡ccnv 4658  dom cdm 4659   “ cima 4662   ∘ ccom 4663  Rel wrel 4664  Fun wfun 5248   Fn wfn 5249  ⟶wf 5250  –1-1-onto→wf1o 5253  ‘cfv 5254  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-ima 4672  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:  omp1eomlem  7153  ctm  7168