Theorem tz6.12f 6910

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 4841	. . . . 5 ⊢ (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
2	1	eleq1d 2850	. . . 4 ⊢ (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3		tz6.12f.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
4	3	nfel2 2945	. . . . . 6 ⊢ Ⅎ𝑦⟨𝐴, 𝑧⟩ ∈ 𝐹
5		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑧⟨𝐴, 𝑦⟩ ∈ 𝐹
6	4, 5, 2	cbveuw 2636	. . . . 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 644	. . 3 ⊢ (𝑧 = 𝑦 → ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)))
9		eqeq2 2777	. . 3 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝐴) = 𝑧 ↔ (𝐹‘𝐴) = 𝑦))
10	8, 9	imbi12d 347	. 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 6909	. 2 ⊢ ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑧)
12	10, 11	chvarvv 2022	1 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃!weu 2598  Ⅎwnfc 2912  ⟨cop 4597  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548

This theorem is used by: (None)