Theorem f1resveqaeq 35228

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 6860	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐶) = (𝐹‘𝐶))
2	1	ad2antrl 729	. . 3 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐶) = (𝐹‘𝐶))
3		fvres 6860	. . . 4 ⊢ (𝐷 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐷) = (𝐹‘𝐷))
4	3	ad2antll 730	. . 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 7211	. 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 1542   ∈ wcel 2114   ↾ 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 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 2709  ax-sep 5232  ax-nul 5242  ax-pr 5376

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fv 6507

This theorem is referenced by:  f1resrcmplf1d  35230