Theorem sectpropd 49695

Description: Two structures with the same base, hom-sets and composition operation have the same sections. (Contributed by Zhi Wang, 27-Oct-2025.)

Hypotheses

Ref	Expression
sectpropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
sectpropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))

Assertion

Ref	Expression
sectpropd	⊢ (𝜑 → (Sect‘𝐶) = (Sect‘𝐷))

Proof of Theorem sectpropd

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sectpropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2		sectpropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
3	1, 2	sectpropdlem 49694	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Sect‘𝐶)) → 𝑓 ∈ (Sect‘𝐷))
4	1	eqcomd 2775	. . . 4 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐶))
5	2	eqcomd 2775	. . . 4 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐶))
6	4, 5	sectpropdlem 49694	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Sect‘𝐷)) → 𝑓 ∈ (Sect‘𝐶))
7	3, 6	impbida 812	. 2 ⊢ (𝜑 → (𝑓 ∈ (Sect‘𝐶) ↔ 𝑓 ∈ (Sect‘𝐷)))
8	7	eqrdv 2767	1 ⊢ (𝜑 → (Sect‘𝐶) = (Sect‘𝐷))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6534  Hom_f chomf 17718  comp_fccomf 17719  Sectcsect 17797

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

This theorem is referenced by:  invpropdlem  49696