isopropdlem - Mathbox for Zhi Wang

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Theorem isopropdlem 49698

Description: Lemma for isopropd 49699. (Contributed by Zhi Wang, 27-Oct-2025.)

**Hypotheses**
Ref	Expression
sectpropd.1	⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
sectpropd.2	⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘𝐷))

**Assertion**
Ref	Expression
isopropdlem	⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐷))

Proof of Theorem isopropdlem Dummy variables 𝑐 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐶))
2		eqid 2769	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2769	. . . . . 6 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		df-iso 17802	. . . . . . . 8 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
5	4	mptrcl 6997	. . . . . . 7 ⊢ (𝑃 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
6	5	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝐶 ∈ Cat)
7		eqid 2769	. . . . . 6 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8	2, 3, 6, 7	isofval2 49690	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Iso‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ dom (𝑥(Inv‘𝐶)𝑦)))
9		df-mpo 7413	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ dom (𝑥(Inv‘𝐶)𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom (𝑥(Inv‘𝐶)𝑦))}
10	8, 9	eqtrdi 2820	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Iso‘𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom (𝑥(Inv‘𝐶)𝑦))})
11	1, 10	eleqtrd 2871	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom (𝑥(Inv‘𝐶)𝑦))})
12		eloprab1st2nd 49526	. . 3 ⊢ (𝑃 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom (𝑥(Inv‘𝐶)𝑦))} → 𝑃 = ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩)
13	11, 12	syl 18	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 = ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩)
14		sectpropd.1	. . . . . . . . 9 ⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
15	14	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Hom_f ‘𝐶) = (Hom_f ‘𝐷))
16		sectpropd.2	. . . . . . . . 9 ⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘𝐷))
17	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (comp_f‘𝐶) = (comp_f‘𝐷))
18	15, 17	invpropd 49697	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Inv‘𝐶) = (Inv‘𝐷))
19	18	oveqd 7425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))) = ((1^st ‘(1^st ‘𝑃))(Inv‘𝐷)(2^nd ‘(1^st ‘𝑃))))
20	19	dmeqd 5893	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐷)(2^nd ‘(1^st ‘𝑃))))
21		eleq1 2857	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑥 ∈ (Base‘𝐶) ↔ (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)))
22	21	anbi1d 642	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ ((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))))
23		oveq1 7415	. . . . . . . . . . 11 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑥(Inv‘𝐶)𝑦) = ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦))
24	23	dmeqd 5893	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → dom (𝑥(Inv‘𝐶)𝑦) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦))
25	24	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑧 = dom (𝑥(Inv‘𝐶)𝑦) ↔ 𝑧 = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦)))
26	22, 25	anbi12d 643	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom (𝑥(Inv‘𝐶)𝑦)) ↔ (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦))))
27		eleq1 2857	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (𝑦 ∈ (Base‘𝐶) ↔ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)))
28	27	anbi2d 641	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ↔ ((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶))))
29		oveq2 7416	. . . . . . . . . . 11 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦) = ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))))
30	29	dmeqd 5893	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))))
31	30	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (𝑧 = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦) ↔ 𝑧 = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃)))))
32	28, 31	anbi12d 643	. . . . . . . 8 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → ((((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦)) ↔ (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ 𝑧 = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))))))
33		eqeq1 2773	. . . . . . . . 9 ⊢ (𝑧 = (2^nd ‘𝑃) → (𝑧 = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))) ↔ (2^nd ‘𝑃) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃)))))
34	33	anbi2d 641	. . . . . . . 8 ⊢ (𝑧 = (2^nd ‘𝑃) → ((((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ 𝑧 = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃)))) ↔ (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ (2^nd ‘𝑃) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))))))
35	26, 32, 34	eloprabi 8056	. . . . . . 7 ⊢ (𝑃 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom (𝑥(Inv‘𝐶)𝑦))} → (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ (2^nd ‘𝑃) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃)))))
36	11, 35	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)) ∧ (2^nd ‘𝑃) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃)))))
37	36	simprd 500	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (2^nd ‘𝑃) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))))
38		eqid 2769	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
39		eqid 2769	. . . . . 6 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
40	36	simplld 779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶))
41	15	homfeqbas 17748	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
42	40, 41	eleqtrd 2871	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐷))
43	42	elfvexd 6915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝐷 ∈ V)
44	15, 17, 6, 43	catpropd 17761	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
45	6, 44	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝐷 ∈ Cat)
46	36	simplrd 781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶))
47	46, 41	eleqtrd 2871	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐷))
48		eqid 2769	. . . . . 6 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
49	38, 39, 45, 42, 47, 48	isoval 17818	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → ((1^st ‘(1^st ‘𝑃))(Iso‘𝐷)(2^nd ‘(1^st ‘𝑃))) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐷)(2^nd ‘(1^st ‘𝑃))))
50	20, 37, 49	3eqtr4rd 2815	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → ((1^st ‘(1^st ‘𝑃))(Iso‘𝐷)(2^nd ‘(1^st ‘𝑃))) = (2^nd ‘𝑃))
51		isofn 17828	. . . . . 6 ⊢ (𝐷 ∈ Cat → (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
52	45, 51	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
53		fnbrovb 7459	. . . . 5 ⊢ (((Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) ∧ ((1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐷) ∧ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐷))) → (((1^st ‘(1^st ‘𝑃))(Iso‘𝐷)(2^nd ‘(1^st ‘𝑃))) = (2^nd ‘𝑃) ↔ ⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Iso‘𝐷)(2^nd ‘𝑃)))
54	52, 42, 47, 53	syl12anc 849	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (((1^st ‘(1^st ‘𝑃))(Iso‘𝐷)(2^nd ‘(1^st ‘𝑃))) = (2^nd ‘𝑃) ↔ ⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Iso‘𝐷)(2^nd ‘𝑃)))
55	50, 54	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → ⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Iso‘𝐷)(2^nd ‘𝑃))
56		df-br 5111	. . 3 ⊢ (⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Iso‘𝐷)(2^nd ‘𝑃) ↔ ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩ ∈ (Iso‘𝐷))
57	55, 56	sylib 221	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩ ∈ (Iso‘𝐷))
58	13, 57	eqeltrd 2869	1 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ⟨cop 4597 class class class wbr 5110 ↦ cmpt 5193 × cxp 5657 dom cdm 5659 ∘ ccom 5663 Fn wfn 6529 ‘cfv 6534 (class class class)co 7408 {coprab 7409 ∈ cmpo 7410 1^st c1st 7980 2^nd c2nd 7981 Basecbs 17265 Catccat 17716 Hom_f chomf 17718 comp_fccomf 17719 Invcinv 17798 Isociso 17799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7982 df-2nd 7983 df-cat 17720 df-cid 17721 df-homf 17722 df-comf 17723 df-sect 17800 df-inv 17801 df-iso 17802

This theorem is referenced by: isopropd 49699

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