Theorem f1ocnvfvrneq 7307

Description: If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Assertion

Ref	Expression
f1ocnvfvrneq	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷))

Proof of Theorem f1ocnvfvrneq

Step	Hyp	Ref	Expression
1		f1f1orn 6858	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
2		f1ocnv 6859	. . 3 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
3		f1of1 6846	. . 3 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹:ran 𝐹–1-1→𝐴)
4		f1veqaeq 7278	. . . 4 ⊢ ((◡𝐹:ran 𝐹–1-1→𝐴 ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷))
5	4	ex 412	. . 3 ⊢ (◡𝐹:ran 𝐹–1-1→𝐴 → ((𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷)))
6	1, 2, 3, 5	4syl 19	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷)))
7	6	imp 406	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ^◡ccnv 5683  ran crn 5685  –1-1→wf1 6557  –1-1-onto→wf1o 6559  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568

This theorem is referenced by: (None)