Theorem f1resveqaeq 35068

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 6859	. . . 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 6859	. . . 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 2745	. 2 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝐶) = ((𝐹 ↾ 𝐴)‘𝐷) ↔ (𝐹‘𝐶) = (𝐹‘𝐷)))
6		f1veqaeq 7213	. 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 5633  –1-1→wf1 6496  ‘cfv 6499

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fv 6507

This theorem is referenced by:  f1resrcmplf1d  35070