Theorem f1resveqaeq 35082

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 6880	. . . 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 6880	. . . 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 2746	. 2 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝐶) = ((𝐹 ↾ 𝐴)‘𝐷) ↔ (𝐹‘𝐶) = (𝐹‘𝐷)))
6		f1veqaeq 7234	. 2 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝐶) = ((𝐹 ↾ 𝐴)‘𝐷) → 𝐶 = 𝐷))
7	5, 6	sylbird 260	1 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ↾ cres 5643  –1-1→wf1 6511  ‘cfv 6514

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fv 6522

This theorem is referenced by:  f1resrcmplf1d  35084