Theorem isopropd 49046

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 49045	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Iso‘𝐶)) → 𝑓 ∈ (Iso‘𝐷))
4	1	eqcomd 2735	. . . 4 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐶))
5	2	eqcomd 2735	. . . 4 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐶))
6	4, 5	isopropdlem 49045	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Iso‘𝐷)) → 𝑓 ∈ (Iso‘𝐶))
7	3, 6	impbida 800	. 2 ⊢ (𝜑 → (𝑓 ∈ (Iso‘𝐶) ↔ 𝑓 ∈ (Iso‘𝐷)))
8	7	eqrdv 2727	1 ⊢ (𝜑 → (Iso‘𝐶) = (Iso‘𝐷))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6486  Hom_f chomf 17591  comp_fccomf 17592  Isociso 17672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-cat 17593  df-cid 17594  df-homf 17595  df-comf 17596  df-sect 17673  df-inv 17674  df-iso 17675

This theorem is referenced by:  cicpropdlem  49054