Theorem tz6.12f 6899

Description: Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)

Hypothesis

Ref	Expression
tz6.12f.1	⊢ Ⅎ𝑦𝐹

Assertion

Ref	Expression
tz6.12f	⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝐹(𝑦)

Proof of Theorem tz6.12f

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4848	. . . . 5 ⊢ (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
2	1	eleq1d 2818	. . . 4 ⊢ (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3		tz6.12f.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
4	3	nfel2 2916	. . . . . 6 ⊢ Ⅎ𝑦⟨𝐴, 𝑧⟩ ∈ 𝐹
5		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑧⟨𝐴, 𝑦⟩ ∈ 𝐹
6	4, 5, 2	cbveuw 2604	. . . . 5 ⊢ (∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
7	6	a1i 11	. . . 4 ⊢ (𝑧 = 𝑦 → (∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹))
8	2, 7	anbi12d 632	. . 3 ⊢ (𝑧 = 𝑦 → ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)))
9		eqeq2 2746	. . 3 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝐴) = 𝑧 ↔ (𝐹‘𝐴) = 𝑦))
10	8, 9	imbi12d 344	. 2 ⊢ (𝑧 = 𝑦 → (((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑧) ↔ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)))
11		tz6.12 6898	. 2 ⊢ ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑧)
12	10, 11	chvarvv 1997	1 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃!weu 2566  Ⅎwnfc 2882  ⟨cop 4605  ‘cfv 6528

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 2706

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 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-iota 6481  df-fv 6536

This theorem is referenced by: (None)