Theorem f1resveqaeq 32770

Description: If a function restricted to a class is one-to-one, then for any two elements of the class, the values of the function at those elements are equal only if the two elements are the same element. (Contributed by BTernaryTau, 27-Sep-2023.)

Assertion

Ref	Expression
f1resveqaeq	⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))

Proof of Theorem f1resveqaeq

Step	Hyp	Ref	Expression
1		fvres 6736	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐶) = (𝐹‘𝐶))
2	1	ad2antrl 728	. . 3 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐶) = (𝐹‘𝐶))
3		fvres 6736	. . . 4 ⊢ (𝐷 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐷) = (𝐹‘𝐷))
4	3	ad2antll 729	. . 3 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐷) = (𝐹‘𝐷))
5	2, 4	eqeq12d 2753	. 2 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝐶) = ((𝐹 ↾ 𝐴)‘𝐷) ↔ (𝐹‘𝐶) = (𝐹‘𝐷)))
6		f1veqaeq 7069	. 2 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝐶) = ((𝐹 ↾ 𝐴)‘𝐷) → 𝐶 = 𝐷))
7	5, 6	sylbird 263	1 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ↾ cres 5553  –1-1→wf1 6377  ‘cfv 6380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-res 5563  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fv 6388

This theorem is referenced by:  f1resrcmplf1d  32772