Theorem caseinl 6862

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 6855	. . . 4 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	fveq1i 5341	. . 3 ⊢ (case(𝐹, 𝐺)‘(inl‘𝐴)) = (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inl‘𝐴))
3		caseinl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐵)
4		fnfun 5145	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
6		djulf1o 6830	. . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V)
7		f1ocnv 5301	. . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → ◡inl:({∅} × V)–1-1-onto→V)
8	6, 7	ax-mp 7	. . . . . . 7 ⊢ ◡inl:({∅} × V)–1-1-onto→V
9		f1ofun 5290	. . . . . . 7 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
10	8, 9	ax-mp 7	. . . . . 6 ⊢ Fun ◡inl
11		funco 5088	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
12	5, 10, 11	sylancl 405	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
13		dmco 4973	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
14		df-inl 6819	. . . . . . . . . . 11 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
15	14	funmpt2 5087	. . . . . . . . . 10 ⊢ Fun inl
16		funrel 5066	. . . . . . . . . 10 ⊢ (Fun inl → Rel inl)
17	15, 16	ax-mp 7	. . . . . . . . 9 ⊢ Rel inl
18		dfrel2 4915	. . . . . . . . 9 ⊢ (Rel inl ↔ ◡◡inl = inl)
19	17, 18	mpbi 144	. . . . . . . 8 ⊢ ◡◡inl = inl
20	19	a1i 9	. . . . . . 7 ⊢ (𝜑 → ◡◡inl = inl)
21		fndm 5147	. . . . . . . 8 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
22	3, 21	syl 14	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐵)
23	20, 22	imaeq12d 4808	. . . . . 6 ⊢ (𝜑 → (◡◡inl “ dom 𝐹) = (inl “ 𝐵))
24	13, 23	syl5eq 2139	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∘ ◡inl) = (inl “ 𝐵))
25		df-fn 5052	. . . . 5 ⊢ ((𝐹 ∘ ◡inl) Fn (inl “ 𝐵) ↔ (Fun (𝐹 ∘ ◡inl) ∧ dom (𝐹 ∘ ◡inl) = (inl “ 𝐵)))
26	12, 24, 25	sylanbrc 409	. . . 4 ⊢ (𝜑 → (𝐹 ∘ ◡inl) Fn (inl “ 𝐵))
27		caseinl.g	. . . . . 6 ⊢ (𝜑 → Fun 𝐺)
28		djurf1o 6831	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
29		f1ocnv 5301	. . . . . . . 8 ⊢ (inr:V–1-1-onto→({1o} × V) → ◡inr:({1o} × V)–1-1-onto→V)
30	28, 29	ax-mp 7	. . . . . . 7 ⊢ ◡inr:({1o} × V)–1-1-onto→V
31		f1ofun 5290	. . . . . . 7 ⊢ (◡inr:({1o} × V)–1-1-onto→V → Fun ◡inr)
32	30, 31	ax-mp 7	. . . . . 6 ⊢ Fun ◡inr
33		funco 5088	. . . . . 6 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
34	27, 32, 33	sylancl 405	. . . . 5 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
35		dmco 4973	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
36		imacnvcnv 4929	. . . . . . 7 ⊢ (◡◡inr “ dom 𝐺) = (inr “ dom 𝐺)
37	35, 36	eqtri 2115	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺)
38	37	a1i 9	. . . . 5 ⊢ (𝜑 → dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺))
39		df-fn 5052	. . . . 5 ⊢ ((𝐺 ∘ ◡inr) Fn (inr “ dom 𝐺) ↔ (Fun (𝐺 ∘ ◡inr) ∧ dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺)))
40	34, 38, 39	sylanbrc 409	. . . 4 ⊢ (𝜑 → (𝐺 ∘ ◡inr) Fn (inr “ dom 𝐺))
41		djuin 6836	. . . . 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 2646	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
45		f1odm 5292	. . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → dom inl = V)
46	6, 45	ax-mp 7	. . . . . . 7 ⊢ dom inl = V
47	44, 46	syl6eleqr 2188	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom inl)
48	47, 15	jctil 306	. . . . 5 ⊢ (𝜑 → (Fun inl ∧ 𝐴 ∈ dom inl))
49		funfvima 5565	. . . . 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 5405	. . . 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 1185	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inl‘𝐴)) = ((𝐹 ∘ ◡inl)‘(inl‘𝐴)))
53	2, 52	syl5eq 2139	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = ((𝐹 ∘ ◡inl)‘(inl‘𝐴)))
54		f1ofn 5289	. . . 4 ⊢ (◡inl:({∅} × V)–1-1-onto→V → ◡inl Fn ({∅} × V))
55	8, 54	ax-mp 7	. . 3 ⊢ ◡inl Fn ({∅} × V)
56		f1of 5288	. . . . . 6 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
57	6, 56	ax-mp 7	. . . . 5 ⊢ inl:V⟶({∅} × V)
58	57	a1i 9	. . . 4 ⊢ (𝜑 → inl:V⟶({∅} × V))
59	58, 44	ffvelrnd 5474	. . 3 ⊢ (𝜑 → (inl‘𝐴) ∈ ({∅} × V))
60		fvco2 5408	. . 3 ⊢ ((◡inl Fn ({∅} × V) ∧ (inl‘𝐴) ∈ ({∅} × V)) → ((𝐹 ∘ ◡inl)‘(inl‘𝐴)) = (𝐹‘(◡inl‘(inl‘𝐴))))
61	55, 59, 60	sylancr 406	. 2 ⊢ (𝜑 → ((𝐹 ∘ ◡inl)‘(inl‘𝐴)) = (𝐹‘(◡inl‘(inl‘𝐴))))
62		f1ocnvfv1 5594	. . . 4 ⊢ ((inl:V–1-1-onto→({∅} × V) ∧ 𝐴 ∈ V) → (◡inl‘(inl‘𝐴)) = 𝐴)
63	6, 44, 62	sylancr 406	. . 3 ⊢ (𝜑 → (◡inl‘(inl‘𝐴)) = 𝐴)
64	63	fveq2d 5344	. 2 ⊢ (𝜑 → (𝐹‘(◡inl‘(inl‘𝐴))) = (𝐹‘𝐴))
65	53, 61, 64	3eqtrd 2131	1 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ∪ cun 3011   ∩ cin 3012  ∅c0 3302  {csn 3466  ⟨cop 3469   × cxp 4465  ^◡ccnv 4466  dom cdm 4467   “ cima 4470   ∘ ccom 4471  Rel wrel 4472  Fun wfun 5043   Fn wfn 5044  ⟶wf 5045  –1-1-onto→wf1o 5048  ‘cfv 5049  1_oc1o 6212  inlcinl 6817  inrcinr 6818  casecdjucase 6854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-inl 6819  df-inr 6820  df-case 6855

This theorem is referenced by:  ctm  6871