Theorem invpropd 49514

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 49513	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Inv‘𝐶)) → 𝑓 ∈ (Inv‘𝐷))
4	1	eqcomd 2742	. . . 4 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐶))
5	2	eqcomd 2742	. . . 4 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐶))
6	4, 5	invpropdlem 49513	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (Inv‘𝐷)) → 𝑓 ∈ (Inv‘𝐶))
7	3, 6	impbida 801	. 2 ⊢ (𝜑 → (𝑓 ∈ (Inv‘𝐶) ↔ 𝑓 ∈ (Inv‘𝐷)))
8	7	eqrdv 2734	1 ⊢ (𝜑 → (Inv‘𝐶) = (Inv‘𝐷))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  Hom_f chomf 17632  comp_fccomf 17633  Invcinv 17712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-cat 17634  df-cid 17635  df-homf 17636  df-comf 17637  df-sect 17714  df-inv 17715

This theorem is referenced by:  isopropdlem  49515