Theorem tz6.12f 5677

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 3868	. . . . 5 ⊢ (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
2	1	eleq1d 2300	. . . 4 ⊢ (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3		tz6.12f.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
4	3	nfel2 2388	. . . . . 6 ⊢ Ⅎ𝑦⟨𝐴, 𝑧⟩ ∈ 𝐹
5		nfv 1577	. . . . . 6 ⊢ Ⅎ𝑧⟨𝐴, 𝑦⟩ ∈ 𝐹
6	4, 5, 2	cbveu 2103	. . . . 5 ⊢ (∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
7	6	a1i 9	. . . 4 ⊢ (𝑧 = 𝑦 → (∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹))
8	2, 7	anbi12d 473	. . 3 ⊢ (𝑧 = 𝑦 → ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)))
9		eqeq2 2241	. . 3 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝐴) = 𝑧 ↔ (𝐹‘𝐴) = 𝑦))
10	8, 9	imbi12d 234	. 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 5676	. 2 ⊢ ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑧)
12	10, 11	chvarv 1990	1 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃!weu 2079   ∈ wcel 2202  Ⅎwnfc 2362  ⟨cop 3676  ‘cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341

This theorem is referenced by: (None)