Theorem f1resveqaeq 7277

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 6904	. . . 4 ⊢ (𝐶 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐶) = (𝐹‘𝐶))
2	1	ad2antrl 741	. . 3 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐶) = (𝐹‘𝐶))
3		fvres 6904	. . . 4 ⊢ (𝐷 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐷) = (𝐹‘𝐷))
4	3	ad2antll 742	. . 3 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐷) = (𝐹‘𝐷))
5	2, 4	eqeq12d 2776	. 2 ⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝐶) = ((𝐹 ↾ 𝐴)‘𝐷) ↔ (𝐹‘𝐶) = (𝐹‘𝐷)))
6		f1veqaeq 7260	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ↾ cres 5652  –1-1→wf1 6535  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546

This theorem is used by:  f1resrcmplf1d  7279