Theorem cicpropd 49211

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

Hypotheses

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

Assertion

Ref	Expression
cicpropd	⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝐷))

Proof of Theorem cicpropd

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cicpropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2		cicpropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
3	1, 2	cicpropdlem 49210	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ ( ≃𝑐 ‘𝐶)) → 𝑓 ∈ ( ≃𝑐 ‘𝐷))
4	1	eqcomd 2739	. . . 4 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐶))
5	2	eqcomd 2739	. . . 4 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐶))
6	4, 5	cicpropdlem 49210	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ ( ≃𝑐 ‘𝐷)) → 𝑓 ∈ ( ≃𝑐 ‘𝐶))
7	3, 6	impbida 800	. 2 ⊢ (𝜑 → (𝑓 ∈ ( ≃𝑐 ‘𝐶) ↔ 𝑓 ∈ ( ≃𝑐 ‘𝐷)))
8	7	eqrdv 2731	1 ⊢ (𝜑 → ( ≃𝑐 ‘𝐶) = ( ≃𝑐 ‘𝐷))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6489  Hom_f chomf 17580  comp_fccomf 17581   ≃_𝑐 ccic 17710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-supp 8100  df-cat 17582  df-cid 17583  df-homf 17584  df-comf 17585  df-sect 17662  df-inv 17663  df-iso 17664  df-cic 17711

This theorem is referenced by:  oppccicb  49212