isopropdlem - Mathbox for Zhi Wang

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Theorem isopropdlem 49515

Description: Lemma for isopropd 49516. (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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐶))
2		eqid 2736	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2736	. . . . . 6 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		df-iso 17716	. . . . . . . 8 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
5	4	mptrcl 6957	. . . . . . 7 ⊢ (𝑃 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
6	5	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝐶 ∈ Cat)
7		eqid 2736	. . . . . 6 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
8	2, 3, 6, 7	isofval2 49507	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Iso‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ dom (𝑥(Inv‘𝐶)𝑦)))
9		df-mpo 7372	. . . . 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 2787	. . . 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 2838	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) ∧ 𝑧 = dom (𝑥(Inv‘𝐶)𝑦))})
12		eloprab1st2nd 49343	. . 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 17	. 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 480	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (comp_f‘𝐶) = (comp_f‘𝐷))
18	15, 17	invpropd 49514	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Inv‘𝐶) = (Inv‘𝐷))
19	18	oveqd 7384	. . . . . 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 5860	. . . . 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 2824	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑥 ∈ (Base‘𝐶) ↔ (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)))
22	21	anbi1d 632	. . . . . . . . 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 7374	. . . . . . . . . . 11 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → (𝑥(Inv‘𝐶)𝑦) = ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦))
24	23	dmeqd 5860	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘(1^st ‘𝑃)) → dom (𝑥(Inv‘𝐶)𝑦) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)𝑦))
25	24	eqeq2d 2747	. . . . . . . . 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 633	. . . . . . . 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 2824	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘(1^st ‘𝑃)) → (𝑦 ∈ (Base‘𝐶) ↔ (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶)))
28	27	anbi2d 631	. . . . . . . . 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 7375	. . . . . . . . . . 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 5860	. . . . . . . . . 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 2747	. . . . . . . . 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 633	. . . . . . . 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 2740	. . . . . . . . 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 631	. . . . . . . 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 8016	. . . . . . 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 17	. . . . . 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 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (2^nd ‘𝑃) = dom ((1^st ‘(1^st ‘𝑃))(Inv‘𝐶)(2^nd ‘(1^st ‘𝑃))))
38		eqid 2736	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
39		eqid 2736	. . . . . 6 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
40	36	simplld 768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐶))
41	15	homfeqbas 17662	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Base‘𝐶) = (Base‘𝐷))
42	40, 41	eleqtrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (1^st ‘(1^st ‘𝑃)) ∈ (Base‘𝐷))
43	42	elfvexd 6876	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝐷 ∈ V)
44	15, 17, 6, 43	catpropd 17675	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
45	6, 44	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝐷 ∈ Cat)
46	36	simplrd 770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐶))
47	46, 41	eleqtrd 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (2^nd ‘(1^st ‘𝑃)) ∈ (Base‘𝐷))
48		eqid 2736	. . . . . 6 ⊢ (Iso‘𝐷) = (Iso‘𝐷)
49	38, 39, 45, 42, 47, 48	isoval 17732	. . . . 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 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → ((1^st ‘(1^st ‘𝑃))(Iso‘𝐷)(2^nd ‘(1^st ‘𝑃))) = (2^nd ‘𝑃))
51		isofn 17742	. . . . . 6 ⊢ (𝐷 ∈ Cat → (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
52	45, 51	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → (Iso‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)))
53		fnbrovb 7418	. . . . 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 837	. . . 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 232	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → ⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩(Iso‘𝐷)(2^nd ‘𝑃))
56		df-br 5086	. . 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 218	. 2 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → ⟨⟨(1^st ‘(1^st ‘𝑃)), (2^nd ‘(1^st ‘𝑃))⟩, (2^nd ‘𝑃)⟩ ∈ (Iso‘𝐷))
58	13, 57	eqeltrd 2836	1 ⊢ ((𝜑 ∧ 𝑃 ∈ (Iso‘𝐶)) → 𝑃 ∈ (Iso‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3429 ⟨cop 4573 class class class wbr 5085 ↦ cmpt 5166 × cxp 5629 dom cdm 5631 ∘ ccom 5635 Fn wfn 6493 ‘cfv 6498 (class class class)co 7367 {coprab 7368 ∈ cmpo 7369 1^st c1st 7940 2^nd c2nd 7941 Basecbs 17179 Catccat 17630 Hom_f chomf 17632 comp_fccomf 17633 Invcinv 17712 Isociso 17713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-cat 17634 df-cid 17635 df-homf 17636 df-comf 17637 df-sect 17714 df-inv 17715 df-iso 17716

This theorem is referenced by: isopropd 49516

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