Theorem caseinl 7192

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 7185	. . . 4 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	fveq1i 5576	. . 3 ⊢ (case(𝐹, 𝐺)‘(inl‘𝐴)) = (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inl‘𝐴))
3		caseinl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐵)
4		fnfun 5370	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
6		djulf1o 7159	. . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V)
7		f1ocnv 5534	. . . . . . . 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 5523	. . . . . . 7 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun ◡inl
11		funco 5310	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
12	5, 10, 11	sylancl 413	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
13		dmco 5190	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
14		df-inl 7148	. . . . . . . . . . 11 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
15	14	funmpt2 5309	. . . . . . . . . 10 ⊢ Fun inl
16		funrel 5287	. . . . . . . . . 10 ⊢ (Fun inl → Rel inl)
17	15, 16	ax-mp 5	. . . . . . . . 9 ⊢ Rel inl
18		dfrel2 5132	. . . . . . . . 9 ⊢ (Rel inl ↔ ◡◡inl = inl)
19	17, 18	mpbi 145	. . . . . . . 8 ⊢ ◡◡inl = inl
20	19	a1i 9	. . . . . . 7 ⊢ (𝜑 → ◡◡inl = inl)
21		fndm 5372	. . . . . . . 8 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
22	3, 21	syl 14	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐵)
23	20, 22	imaeq12d 5022	. . . . . 6 ⊢ (𝜑 → (◡◡inl “ dom 𝐹) = (inl “ 𝐵))
24	13, 23	eqtrid 2249	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∘ ◡inl) = (inl “ 𝐵))
25		df-fn 5273	. . . . 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 7160	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
29		f1ocnv 5534	. . . . . . . 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 5523	. . . . . . 7 ⊢ (◡inr:({1o} × V)–1-1-onto→V → Fun ◡inr)
32	30, 31	ax-mp 5	. . . . . 6 ⊢ Fun ◡inr
33		funco 5310	. . . . . 6 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
34	27, 32, 33	sylancl 413	. . . . 5 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
35		dmco 5190	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
36		imacnvcnv 5146	. . . . . . 7 ⊢ (◡◡inr “ dom 𝐺) = (inr “ dom 𝐺)
37	35, 36	eqtri 2225	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺)
38	37	a1i 9	. . . . 5 ⊢ (𝜑 → dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺))
39		df-fn 5273	. . . . 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 7165	. . . . 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 2784	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
45		f1odm 5525	. . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → dom inl = V)
46	6, 45	ax-mp 5	. . . . . . 7 ⊢ dom inl = V
47	44, 46	eleqtrrdi 2298	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom inl)
48	47, 15	jctil 312	. . . . 5 ⊢ (𝜑 → (Fun inl ∧ 𝐴 ∈ dom inl))
49		funfvima 5815	. . . . 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 5644	. . . 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 2249	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = ((𝐹 ∘ ◡inl)‘(inl‘𝐴)))
54		f1ofn 5522	. . . 4 ⊢ (◡inl:({∅} × V)–1-1-onto→V → ◡inl Fn ({∅} × V))
55	8, 54	ax-mp 5	. . 3 ⊢ ◡inl Fn ({∅} × V)
56		f1of 5521	. . . . . 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 5715	. . 3 ⊢ (𝜑 → (inl‘𝐴) ∈ ({∅} × V))
60		fvco2 5647	. . 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 5845	. . . 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 5579	. 2 ⊢ (𝜑 → (𝐹‘(◡inl‘(inl‘𝐴))) = (𝐹‘𝐴))
65	53, 61, 64	3eqtrd 2241	1 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∪ cun 3163   ∩ cin 3164  ∅c0 3459  {csn 3632  ⟨cop 3635   × cxp 4672  ^◡ccnv 4673  dom cdm 4674   “ cima 4677   ∘ ccom 4678  Rel wrel 4679  Fun wfun 5264   Fn wfn 5265  ⟶wf 5266  –1-1-onto→wf1o 5269  ‘cfv 5270  1_oc1o 6494  inlcinl 7146  inrcinr 7147  casecdjucase 7184

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1st 6225  df-2nd 6226  df-1o 6501  df-inl 7148  df-inr 7149  df-case 7185

This theorem is referenced by:  omp1eomlem  7195  ctm  7210