Theorem caseinl 7421

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 7414	. . . 4 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))
2	1	fveq1i 5691	. . 3 ⊢ (case(𝐹, 𝐺)‘(inl‘𝐴)) = (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inl‘𝐴))
3		caseinl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐵)
4		fnfun 5473	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
6		djulf1o 7388	. . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V)
7		f1ocnv 5647	. . . . . . . 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 5636	. . . . . . 7 ⊢ (◡inl:({∅} × V)–1-1-onto→V → Fun ◡inl)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun ◡inl
11		funco 5412	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun ◡inl) → Fun (𝐹 ∘ ◡inl))
12	5, 10, 11	sylancl 417	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ∘ ◡inl))
13		dmco 5291	. . . . . 6 ⊢ dom (𝐹 ∘ ◡inl) = (◡◡inl “ dom 𝐹)
14		df-inl 7377	. . . . . . . . . . 11 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
15	14	funmpt2 5411	. . . . . . . . . 10 ⊢ Fun inl
16		funrel 5389	. . . . . . . . . 10 ⊢ (Fun inl → Rel inl)
17	15, 16	ax-mp 5	. . . . . . . . 9 ⊢ Rel inl
18		dfrel2 5233	. . . . . . . . 9 ⊢ (Rel inl ↔ ◡◡inl = inl)
19	17, 18	mpbi 145	. . . . . . . 8 ⊢ ◡◡inl = inl
20	19	a1i 9	. . . . . . 7 ⊢ (𝜑 → ◡◡inl = inl)
21		fndm 5475	. . . . . . . 8 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
22	3, 21	syl 14	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐵)
23	20, 22	imaeq12d 5122	. . . . . 6 ⊢ (𝜑 → (◡◡inl “ dom 𝐹) = (inl “ 𝐵))
24	13, 23	eqtrid 2283	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∘ ◡inl) = (inl “ 𝐵))
25		df-fn 5375	. . . . 5 ⊢ ((𝐹 ∘ ◡inl) Fn (inl “ 𝐵) ↔ (Fun (𝐹 ∘ ◡inl) ∧ dom (𝐹 ∘ ◡inl) = (inl “ 𝐵)))
26	12, 24, 25	sylanbrc 421	. . . 4 ⊢ (𝜑 → (𝐹 ∘ ◡inl) Fn (inl “ 𝐵))
27		caseinl.g	. . . . . 6 ⊢ (𝜑 → Fun 𝐺)
28		djurf1o 7389	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V)
29		f1ocnv 5647	. . . . . . . 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 5636	. . . . . . 7 ⊢ (◡inr:({1o} × V)–1-1-onto→V → Fun ◡inr)
32	30, 31	ax-mp 5	. . . . . 6 ⊢ Fun ◡inr
33		funco 5412	. . . . . 6 ⊢ ((Fun 𝐺 ∧ Fun ◡inr) → Fun (𝐺 ∘ ◡inr))
34	27, 32, 33	sylancl 417	. . . . 5 ⊢ (𝜑 → Fun (𝐺 ∘ ◡inr))
35		dmco 5291	. . . . . . 7 ⊢ dom (𝐺 ∘ ◡inr) = (◡◡inr “ dom 𝐺)
36		imacnvcnv 5247	. . . . . . 7 ⊢ (◡◡inr “ dom 𝐺) = (inr “ dom 𝐺)
37	35, 36	eqtri 2259	. . . . . 6 ⊢ dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺)
38	37	a1i 9	. . . . 5 ⊢ (𝜑 → dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺))
39		df-fn 5375	. . . . 5 ⊢ ((𝐺 ∘ ◡inr) Fn (inr “ dom 𝐺) ↔ (Fun (𝐺 ∘ ◡inr) ∧ dom (𝐺 ∘ ◡inr) = (inr “ dom 𝐺)))
40	34, 38, 39	sylanbrc 421	. . . 4 ⊢ (𝜑 → (𝐺 ∘ ◡inr) Fn (inr “ dom 𝐺))
41		djuin 7394	. . . . 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 2835	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
45		f1odm 5638	. . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → dom inl = V)
46	6, 45	ax-mp 5	. . . . . . 7 ⊢ dom inl = V
47	44, 46	eleqtrrdi 2332	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom inl)
48	47, 15	jctil 312	. . . . 5 ⊢ (𝜑 → (Fun inl ∧ 𝐴 ∈ dom inl))
49		funfvima 5940	. . . . 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 5763	. . . 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 1282	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ◡inl) ∪ (𝐺 ∘ ◡inr))‘(inl‘𝐴)) = ((𝐹 ∘ ◡inl)‘(inl‘𝐴)))
53	2, 52	eqtrid 2283	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = ((𝐹 ∘ ◡inl)‘(inl‘𝐴)))
54		f1ofn 5635	. . . 4 ⊢ (◡inl:({∅} × V)–1-1-onto→V → ◡inl Fn ({∅} × V))
55	8, 54	ax-mp 5	. . 3 ⊢ ◡inl Fn ({∅} × V)
56		f1of 5634	. . . . . 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 5835	. . 3 ⊢ (𝜑 → (inl‘𝐴) ∈ ({∅} × V))
60		fvco2 5768	. . 3 ⊢ ((◡inl Fn ({∅} × V) ∧ (inl‘𝐴) ∈ ({∅} × V)) → ((𝐹 ∘ ◡inl)‘(inl‘𝐴)) = (𝐹‘(◡inl‘(inl‘𝐴))))
61	55, 59, 60	sylancr 418	. 2 ⊢ (𝜑 → ((𝐹 ∘ ◡inl)‘(inl‘𝐴)) = (𝐹‘(◡inl‘(inl‘𝐴))))
62		f1ocnvfv1 5973	. . . 4 ⊢ ((inl:V–1-1-onto→({∅} × V) ∧ 𝐴 ∈ V) → (◡inl‘(inl‘𝐴)) = 𝐴)
63	6, 44, 62	sylancr 418	. . 3 ⊢ (𝜑 → (◡inl‘(inl‘𝐴)) = 𝐴)
64	63	fveq2d 5694	. 2 ⊢ (𝜑 → (𝐹‘(◡inl‘(inl‘𝐴))) = (𝐹‘𝐴))
65	53, 61, 64	3eqtrd 2275	1 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218   ∩ cin 3219  ∅c0 3520  {csn 3705  ⟨cop 3708   × cxp 4767  ^◡ccnv 4768  dom cdm 4769   “ cima 4772   ∘ ccom 4773  Rel wrel 4774  Fun wfun 5366   Fn wfn 5367  ⟶wf 5368  –1-1-onto→wf1o 5371  ‘cfv 5372  1_oc1o 6670  inlcinl 7375  inrcinr 7376  casecdjucase 7413

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-inl 7377  df-inr 7378  df-case 7414

This theorem is referenced by:  omp1eomlem  7424  ctm  7439