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Theorem caseinl 7090

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 7083	. . . 4 ⊢ case(𝐹, 𝐺) = ((𝐹 ∘ ^◡inl) ∪ (𝐺 ∘ ^◡inr))
2	1	fveq1i 5517	. . 3 ⊢ (case(𝐹, 𝐺)‘(inl‘𝐴)) = (((𝐹 ∘ ^◡inl) ∪ (𝐺 ∘ ^◡inr))‘(inl‘𝐴))
3		caseinl.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐵)
4		fnfun 5314	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝐹)
6		djulf1o 7057	. . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V)
7		f1ocnv 5475	. . . . . . . 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 5464	. . . . . . 7 ⊢ (^◡inl:({∅} × V)–1-1-onto→V → Fun ^◡inl)
10	8, 9	ax-mp 5	. . . . . 6 ⊢ Fun ^◡inl
11		funco 5257	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun ^◡inl) → Fun (𝐹 ∘ ^◡inl))
12	5, 10, 11	sylancl 413	. . . . 5 ⊢ (𝜑 → Fun (𝐹 ∘ ^◡inl))
13		dmco 5138	. . . . . 6 ⊢ dom (𝐹 ∘ ^◡inl) = (^◡^◡inl “ dom 𝐹)
14		df-inl 7046	. . . . . . . . . . 11 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
15	14	funmpt2 5256	. . . . . . . . . 10 ⊢ Fun inl
16		funrel 5234	. . . . . . . . . 10 ⊢ (Fun inl → Rel inl)
17	15, 16	ax-mp 5	. . . . . . . . 9 ⊢ Rel inl
18		dfrel2 5080	. . . . . . . . 9 ⊢ (Rel inl ↔ ^◡^◡inl = inl)
19	17, 18	mpbi 145	. . . . . . . 8 ⊢ ^◡^◡inl = inl
20	19	a1i 9	. . . . . . 7 ⊢ (𝜑 → ^◡^◡inl = inl)
21		fndm 5316	. . . . . . . 8 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
22	3, 21	syl 14	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐵)
23	20, 22	imaeq12d 4972	. . . . . 6 ⊢ (𝜑 → (^◡^◡inl “ dom 𝐹) = (inl “ 𝐵))
24	13, 23	eqtrid 2222	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∘ ^◡inl) = (inl “ 𝐵))
25		df-fn 5220	. . . . 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 7058	. . . . . . . 8 ⊢ inr:V–1-1-onto→({1_o} × V)
29		f1ocnv 5475	. . . . . . . 8 ⊢ (inr:V–1-1-onto→({1_o} × V) → ^◡inr:({1_o} × V)–1-1-onto→V)
30	28, 29	ax-mp 5	. . . . . . 7 ⊢ ^◡inr:({1_o} × V)–1-1-onto→V
31		f1ofun 5464	. . . . . . 7 ⊢ (^◡inr:({1_o} × V)–1-1-onto→V → Fun ^◡inr)
32	30, 31	ax-mp 5	. . . . . 6 ⊢ Fun ^◡inr
33		funco 5257	. . . . . 6 ⊢ ((Fun 𝐺 ∧ Fun ^◡inr) → Fun (𝐺 ∘ ^◡inr))
34	27, 32, 33	sylancl 413	. . . . 5 ⊢ (𝜑 → Fun (𝐺 ∘ ^◡inr))
35		dmco 5138	. . . . . . 7 ⊢ dom (𝐺 ∘ ^◡inr) = (^◡^◡inr “ dom 𝐺)
36		imacnvcnv 5094	. . . . . . 7 ⊢ (^◡^◡inr “ dom 𝐺) = (inr “ dom 𝐺)
37	35, 36	eqtri 2198	. . . . . 6 ⊢ dom (𝐺 ∘ ^◡inr) = (inr “ dom 𝐺)
38	37	a1i 9	. . . . 5 ⊢ (𝜑 → dom (𝐺 ∘ ^◡inr) = (inr “ dom 𝐺))
39		df-fn 5220	. . . . 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 7063	. . . . 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 2751	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
45		f1odm 5466	. . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → dom inl = V)
46	6, 45	ax-mp 5	. . . . . . 7 ⊢ dom inl = V
47	44, 46	eleqtrrdi 2271	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom inl)
48	47, 15	jctil 312	. . . . 5 ⊢ (𝜑 → (Fun inl ∧ 𝐴 ∈ dom inl))
49		funfvima 5749	. . . . 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 5583	. . . 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 1242	. . 3 ⊢ (𝜑 → (((𝐹 ∘ ^◡inl) ∪ (𝐺 ∘ ^◡inr))‘(inl‘𝐴)) = ((𝐹 ∘ ^◡inl)‘(inl‘𝐴)))
53	2, 52	eqtrid 2222	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = ((𝐹 ∘ ^◡inl)‘(inl‘𝐴)))
54		f1ofn 5463	. . . 4 ⊢ (^◡inl:({∅} × V)–1-1-onto→V → ^◡inl Fn ({∅} × V))
55	8, 54	ax-mp 5	. . 3 ⊢ ^◡inl Fn ({∅} × V)
56		f1of 5462	. . . . . 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 5653	. . 3 ⊢ (𝜑 → (inl‘𝐴) ∈ ({∅} × V))
60		fvco2 5586	. . 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 5778	. . . 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 5520	. 2 ⊢ (𝜑 → (𝐹‘(^◡inl‘(inl‘𝐴))) = (𝐹‘𝐴))
65	53, 61, 64	3eqtrd 2214	1 ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹‘𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2738 ∪ cun 3128 ∩ cin 3129 ∅c0 3423 {csn 3593 ⟨cop 3596 × cxp 4625 ^◡ccnv 4626 dom cdm 4627 “ cima 4630 ∘ ccom 4631 Rel wrel 4632 Fun wfun 5211 Fn wfn 5212 ⟶wf 5213 –1-1-onto→wf1o 5216 ‘cfv 5217 1_oc1o 6410 inlcinl 7044 inrcinr 7045 casecdjucase 7082

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-iord 4367 df-on 4369 df-suc 4372 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-1st 6141 df-2nd 6142 df-1o 6417 df-inl 7046 df-inr 7047 df-case 7083

This theorem is referenced by: omp1eomlem 7093 ctm 7108

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