Theorem isopropd 48915

Description: Two structures with the same base, hom-sets and composition operation have the same isomorphisms. (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
isopropd	⊢ (𝜑 → (Iso‘𝐶) = (Iso‘𝐷))

Proof of Theorem isopropd

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	isopropdlem 48914	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Iso‘𝐶)) → 𝑓 ∈ (Iso‘𝐷))
4	1	eqcomd 2740	. . . 4 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐶))
5	2	eqcomd 2740	. . . 4 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐶))
6	4, 5	isopropdlem 48914	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Iso‘𝐷)) → 𝑓 ∈ (Iso‘𝐶))
7	3, 6	impbida 800	. 2 ⊢ (𝜑 → (𝑓 ∈ (Iso‘𝐶) ↔ 𝑓 ∈ (Iso‘𝐷)))
8	7	eqrdv 2732	1 ⊢ (𝜑 → (Iso‘𝐶) = (Iso‘𝐷))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6541  Hom_f chomf 17681  comp_fccomf 17682  Isociso 17762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-cat 17683  df-cid 17684  df-homf 17685  df-comf 17686  df-sect 17763  df-inv 17764  df-iso 17765

This theorem is referenced by:  cicpropdlem  48923