Theorem caseinr 7351

Description: Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)

Hypotheses

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

Assertion

Ref	Expression
caseinr	⊢ (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺‘𝐴))

Proof of Theorem caseinr

Step	Hyp	Ref	Expression
1		df-case 7343	. . . 4 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	fveq1i 5649	. . 3 ⊢ (case(𝐹, 𝐺)‘(inr‘𝐴)) = (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inr‘𝐴))
3		caseinr.f	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
4		djulf1o 7317	. . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V)
5		f1ocnv 5605	. . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → ◡inl:({∅} × V)–1-1-onto→V)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ ◡inl:({∅} × V)–1-1-onto→V
7		f1ofun 5594	. . . . . . 7 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
8	6, 7	ax-mp 5	. . . . . 6 ⊢ Fun ◡inl
9		funco 5373	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
10	3, 8, 9	sylancl 413	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
11		dmco 5252	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
12		imacnvcnv 5208	. . . . . . 7 ⊢ (◡◡inl “ dom 𝐹) = (inl “ dom 𝐹)
13	11, 12	eqtri 2252	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) = (inl “ dom 𝐹)
14	13	a1i 9	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∘ ◡inl) = (inl “ dom 𝐹))
15		df-fn 5336	. . . . 5 ⊢ ((𝐹 ∘ ◡inl) Fn (inl “ dom 𝐹) ↔ (Fun (𝐹 ∘ ◡inl) ∧ dom (𝐹 ∘ ◡inl) = (inl “ dom 𝐹)))
16	10, 14, 15	sylanbrc 417	. . . 4 ⊢ (𝜑 → (𝐹 ∘ ◡inl) Fn (inl “ dom 𝐹))
17		caseinr.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐵)
18		fnfun 5434	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
19	17, 18	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐺)
20		djurf1o 7318	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
21		f1ocnv 5605	. . . . . . . 8 ⊢ (inr:V–1-1-onto→({1o} × V) → ◡inr:({1o} × V)–1-1-onto→V)
22	20, 21	ax-mp 5	. . . . . . 7 ⊢ ◡inr:({1o} × V)–1-1-onto→V
23		f1ofun 5594	. . . . . . 7 ⊢ (◡inr:({1o} × V)–1-1-onto→V → Fun ◡inr)
24	22, 23	ax-mp 5	. . . . . 6 ⊢ Fun ◡inr
25		funco 5373	. . . . . 6 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
26	19, 24, 25	sylancl 413	. . . . 5 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
27		dmco 5252	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
28		df-inr 7307	. . . . . . . . . . 11 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
29	28	funmpt2 5372	. . . . . . . . . 10 ⊢ Fun inr
30		funrel 5350	. . . . . . . . . 10 ⊢ (Fun inr → Rel inr)
31	29, 30	ax-mp 5	. . . . . . . . 9 ⊢ Rel inr
32		dfrel2 5194	. . . . . . . . 9 ⊢ (Rel inr ↔ ◡◡inr = inr)
33	31, 32	mpbi 145	. . . . . . . 8 ⊢ ◡◡inr = inr
34	33	a1i 9	. . . . . . 7 ⊢ (𝜑 → ◡◡inr = inr)
35		fndm 5436	. . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
36	17, 35	syl 14	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝐵)
37	34, 36	imaeq12d 5083	. . . . . 6 ⊢ (𝜑 → (◡◡inr “ dom 𝐺) = (inr “ 𝐵))
38	27, 37	eqtrid 2276	. . . . 5 ⊢ (𝜑 → dom (𝐺 ∘ ◡inr) = (inr “ 𝐵))
39		df-fn 5336	. . . . 5 ⊢ ((𝐺 ∘ ◡inr) Fn (inr “ 𝐵) ↔ (Fun (𝐺 ∘ ◡inr) ∧ dom (𝐺 ∘ ◡inr) = (inr “ 𝐵)))
40	26, 38, 39	sylanbrc 417	. . . 4 ⊢ (𝜑 → (𝐺 ∘ ◡inr) Fn (inr “ 𝐵))
41		djuin 7323	. . . . 5 ⊢ ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅
42	41	a1i 9	. . . 4 ⊢ (𝜑 → ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅)
43		caseinr.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
44	43	elexd 2817	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
45		f1odm 5596	. . . . . . . 8 ⊢ (inr:V–1-1-onto→({1o} × V) → dom inr = V)
46	20, 45	ax-mp 5	. . . . . . 7 ⊢ dom inr = V
47	44, 46	eleqtrrdi 2325	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom inr)
48	47, 29	jctil 312	. . . . 5 ⊢ (𝜑 → (Fun inr ∧ 𝐴 ∈ dom inr))
49		funfvima 5896	. . . . 5 ⊢ ((Fun inr ∧ 𝐴 ∈ dom inr) → (𝐴 ∈ 𝐵 → (inr‘𝐴) ∈ (inr “ 𝐵)))
50	48, 43, 49	sylc 62	. . . 4 ⊢ (𝜑 → (inr‘𝐴) ∈ (inr “ 𝐵))
51		fvun2 5722	. . . 4 ⊢ (((𝐹 ∘ ◡inl) Fn (inl “ dom 𝐹) ∧ (𝐺 ∘ ◡inr) Fn (inr “ 𝐵) ∧ (((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅ ∧ (inr‘𝐴) ∈ (inr “ 𝐵))) → (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inr‘𝐴)) = ((𝐺 ∘ ◡inr)‘(inr‘𝐴)))
52	16, 40, 42, 50, 51	syl112anc 1278	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inr‘𝐴)) = ((𝐺 ∘ ◡inr)‘(inr‘𝐴)))
53	2, 52	eqtrid 2276	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = ((𝐺 ∘ ◡inr)‘(inr‘𝐴)))
54		f1ofn 5593	. . . 4 ⊢ (◡inr:({1o} × V)–1-1-onto→V → ◡inr Fn ({1o} × V))
55	22, 54	ax-mp 5	. . 3 ⊢ ◡inr Fn ({1o} × V)
56		f1of 5592	. . . . . 6 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
57	20, 56	ax-mp 5	. . . . 5 ⊢ inr:V⟶({1o} × V)
58	57	a1i 9	. . . 4 ⊢ (𝜑 → inr:V⟶({1o} × V))
59	58, 44	ffvelcdmd 5791	. . 3 ⊢ (𝜑 → (inr‘𝐴) ∈ ({1o} × V))
60		fvco2 5724	. . 3 ⊢ ((◡inr Fn ({1o} × V) ∧ (inr‘𝐴) ∈ ({1o} × V)) → ((𝐺 ∘ ◡inr)‘(inr‘𝐴)) = (𝐺‘(◡inr‘(inr‘𝐴))))
61	55, 59, 60	sylancr 414	. 2 ⊢ (𝜑 → ((𝐺 ∘ ◡inr)‘(inr‘𝐴)) = (𝐺‘(◡inr‘(inr‘𝐴))))
62		f1ocnvfv1 5928	. . . 4 ⊢ ((inr:V–1-1-onto→({1o} × V) ∧ 𝐴 ∈ V) → (◡inr‘(inr‘𝐴)) = 𝐴)
63	20, 44, 62	sylancr 414	. . 3 ⊢ (𝜑 → (◡inr‘(inr‘𝐴)) = 𝐴)
64	63	fveq2d 5652	. 2 ⊢ (𝜑 → (𝐺‘(◡inr‘(inr‘𝐴))) = (𝐺‘𝐴))
65	53, 61, 64	3eqtrd 2268	1 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∪ cun 3199   ∩ cin 3200  ∅c0 3496  {csn 3673  ⟨cop 3676   × cxp 4729  ^◡ccnv 4730  dom cdm 4731   “ cima 4734   ∘ ccom 4735  Rel wrel 4736  Fun wfun 5327   Fn wfn 5328  ⟶wf 5329  –1-1-onto→wf1o 5332  ‘cfv 5333  1_oc1o 6618  inlcinl 7304  inrcinr 7305  casecdjucase 7342

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-inl 7306  df-inr 7307  df-case 7343

This theorem is referenced by:  omp1eomlem  7353