Theorem tz6.12f 5719

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 3900	. . . . 5 ⊢ (𝑧 = 𝑦 → ⟨𝐴, 𝑧⟩ = ⟨𝐴, 𝑦⟩)
2	1	eleq1d 2307	. . . 4 ⊢ (𝑧 = 𝑦 → (⟨𝐴, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝑦⟩ ∈ 𝐹))
3		tz6.12f.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐹
4	3	nfel2 2405	. . . . . 6 ⊢ Ⅎ𝑦⟨𝐴, 𝑧⟩ ∈ 𝐹
5		nfv 1581	. . . . . 6 ⊢ Ⅎ𝑧⟨𝐴, 𝑦⟩ ∈ 𝐹
6	4, 5, 2	cbveu 2110	. . . . 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 477	. . 3 ⊢ (𝑧 = 𝑦 → ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)))
9		eqeq2 2248	. . 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 5718	. 2 ⊢ ((⟨𝐴, 𝑧⟩ ∈ 𝐹 ∧ ∃!𝑧⟨𝐴, 𝑧⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑧)
12	10, 11	chvarv 1997	1 ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃!weu 2086   ∈ wcel 2209  Ⅎwnfc 2379  ⟨cop 3708  ‘cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by: (None)