Theorem invpropd 50260

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

Proof of Theorem invpropd

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	invpropdlem 50259	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Inv‘𝐶)) → 𝑓 ∈ (Inv‘𝐷))
4	1	eqcomd 2766	. . . 4 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐶))
5	2	eqcomd 2766	. . . 4 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐶))
6	4, 5	invpropdlem 50259	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Inv‘𝐷)) → 𝑓 ∈ (Inv‘𝐶))
7	3, 6	impbida 813	. 2 ⊢ (𝜑 → (𝑓 ∈ (Inv‘𝐶) ↔ 𝑓 ∈ (Inv‘𝐷)))
8	7	eqrdv 2758	1 ⊢ (𝜑 → (Inv‘𝐶) = (Inv‘𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6541  Hom_f chomf 17847  comp_fccomf 17848  Invcinv 17927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-cat 17849  df-cid 17850  df-homf 17851  df-comf 17852  df-sect 17929  df-inv 17930

This theorem is used by:  isopropdlem  50261