Theorem caseinr 7048

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 7040	. . . 4 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	fveq1i 5481	. . 3 ⊢ (case(𝐹, 𝐺)‘(inr‘𝐴)) = (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inr‘𝐴))
3		caseinr.f	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
4		djulf1o 7014	. . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V)
5		f1ocnv 5439	. . . . . . . 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 5428	. . . . . . 7 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
8	6, 7	ax-mp 5	. . . . . 6 ⊢ Fun ◡inl
9		funco 5222	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
10	3, 8, 9	sylancl 410	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
11		dmco 5106	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
12		imacnvcnv 5062	. . . . . . 7 ⊢ (◡◡inl “ dom 𝐹) = (inl “ dom 𝐹)
13	11, 12	eqtri 2185	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) = (inl “ dom 𝐹)
14	13	a1i 9	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∘ ◡inl) = (inl “ dom 𝐹))
15		df-fn 5185	. . . . 5 ⊢ ((𝐹 ∘ ◡inl) Fn (inl “ dom 𝐹) ↔ (Fun (𝐹 ∘ ◡inl) ∧ dom (𝐹 ∘ ◡inl) = (inl “ dom 𝐹)))
16	10, 14, 15	sylanbrc 414	. . . 4 ⊢ (𝜑 → (𝐹 ∘ ◡inl) Fn (inl “ dom 𝐹))
17		caseinr.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝐵)
18		fnfun 5279	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
19	17, 18	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐺)
20		djurf1o 7015	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
21		f1ocnv 5439	. . . . . . . 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 5428	. . . . . . 7 ⊢ (◡inr:({1o} × V)–1-1-onto→V → Fun ◡inr)
24	22, 23	ax-mp 5	. . . . . 6 ⊢ Fun ◡inr
25		funco 5222	. . . . . 6 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
26	19, 24, 25	sylancl 410	. . . . 5 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
27		dmco 5106	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
28		df-inr 7004	. . . . . . . . . . 11 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
29	28	funmpt2 5221	. . . . . . . . . 10 ⊢ Fun inr
30		funrel 5199	. . . . . . . . . 10 ⊢ (Fun inr → Rel inr)
31	29, 30	ax-mp 5	. . . . . . . . 9 ⊢ Rel inr
32		dfrel2 5048	. . . . . . . . 9 ⊢ (Rel inr ↔ ◡◡inr = inr)
33	31, 32	mpbi 144	. . . . . . . 8 ⊢ ◡◡inr = inr
34	33	a1i 9	. . . . . . 7 ⊢ (𝜑 → ◡◡inr = inr)
35		fndm 5281	. . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
36	17, 35	syl 14	. . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝐵)
37	34, 36	imaeq12d 4941	. . . . . 6 ⊢ (𝜑 → (◡◡inr “ dom 𝐺) = (inr “ 𝐵))
38	27, 37	syl5eq 2209	. . . . 5 ⊢ (𝜑 → dom (𝐺 ∘ ◡inr) = (inr “ 𝐵))
39		df-fn 5185	. . . . 5 ⊢ ((𝐺 ∘ ◡inr) Fn (inr “ 𝐵) ↔ (Fun (𝐺 ∘ ◡inr) ∧ dom (𝐺 ∘ ◡inr) = (inr “ 𝐵)))
40	26, 38, 39	sylanbrc 414	. . . 4 ⊢ (𝜑 → (𝐺 ∘ ◡inr) Fn (inr “ 𝐵))
41		djuin 7020	. . . . 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 2734	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
45		f1odm 5430	. . . . . . . 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 2258	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom inr)
48	47, 29	jctil 310	. . . . 5 ⊢ (𝜑 → (Fun inr ∧ 𝐴 ∈ dom inr))
49		funfvima 5710	. . . . 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 5547	. . . 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 1231	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inr‘𝐴)) = ((𝐺 ∘ ◡inr)‘(inr‘𝐴)))
53	2, 52	syl5eq 2209	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = ((𝐺 ∘ ◡inr)‘(inr‘𝐴)))
54		f1ofn 5427	. . . 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 5426	. . . . . 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	ffvelrnd 5615	. . 3 ⊢ (𝜑 → (inr‘𝐴) ∈ ({1o} × V))
60		fvco2 5549	. . 3 ⊢ ((◡inr Fn ({1o} × V) ∧ (inr‘𝐴) ∈ ({1o} × V)) → ((𝐺 ∘ ◡inr)‘(inr‘𝐴)) = (𝐺‘(◡inr‘(inr‘𝐴))))
61	55, 59, 60	sylancr 411	. 2 ⊢ (𝜑 → ((𝐺 ∘ ◡inr)‘(inr‘𝐴)) = (𝐺‘(◡inr‘(inr‘𝐴))))
62		f1ocnvfv1 5739	. . . 4 ⊢ ((inr:V–1-1-onto→({1o} × V) ∧ 𝐴 ∈ V) → (◡inr‘(inr‘𝐴)) = 𝐴)
63	20, 44, 62	sylancr 411	. . 3 ⊢ (𝜑 → (◡inr‘(inr‘𝐴)) = 𝐴)
64	63	fveq2d 5484	. 2 ⊢ (𝜑 → (𝐺‘(◡inr‘(inr‘𝐴))) = (𝐺‘𝐴))
65	53, 61, 64	3eqtrd 2201	1 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺‘𝐴))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1342   ∈ wcel 2135  Vcvv 2721   ∪ cun 3109   ∩ cin 3110  ∅c0 3404  {csn 3570  ⟨cop 3573   × cxp 4596  ^◡ccnv 4597  dom cdm 4598   “ cima 4601   ∘ ccom 4602  Rel wrel 4603  Fun wfun 5176   Fn wfn 5177  ⟶wf 5178  –1-1-onto→wf1o 5181  ‘cfv 5182  1_oc1o 6368  inlcinl 7001  inrcinr 7002  casecdjucase 7039

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-suc 4343  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-1st 6100  df-2nd 6101  df-1o 6375  df-inl 7003  df-inr 7004  df-case 7040

This theorem is referenced by:  omp1eomlem  7050