Theorem sectpropd 49527

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 49526	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Sect‘𝐶)) → 𝑓 ∈ (Sect‘𝐷))
4	1	eqcomd 2745	. . . 4 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐶))
5	2	eqcomd 2745	. . . 4 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐶))
6	4, 5	sectpropdlem 49526	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Sect‘𝐷)) → 𝑓 ∈ (Sect‘𝐶))
7	3, 6	impbida 806	. 2 ⊢ (𝜑 → (𝑓 ∈ (Sect‘𝐶) ↔ 𝑓 ∈ (Sect‘𝐷)))
8	7	eqrdv 2737	1 ⊢ (𝜑 → (Sect‘𝐶) = (Sect‘𝐷))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  Hom_f chomf 17623  comp_fccomf 17624  Sectcsect 17702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-cat 17625  df-cid 17626  df-homf 17627  df-comf 17628  df-sect 17705

This theorem is referenced by:  invpropdlem  49528