Theorem nfmpt2 6724

Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Hypotheses

Ref	Expression
nfmpt2.1	⊢ Ⅎ𝑧𝐴
nfmpt2.2	⊢ Ⅎ𝑧𝐵
nfmpt2.3	⊢ Ⅎ𝑧𝐶

Assertion

Ref	Expression
nfmpt2	⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem nfmpt2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt2 6655	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		nfmpt2.1	. . . . . 6 ⊢ Ⅎ𝑧𝐴
3	2	nfcri 2758	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
4		nfmpt2.2	. . . . . 6 ⊢ Ⅎ𝑧𝐵
5	4	nfcri 2758	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
6	3, 5	nfan 1828	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7		nfmpt2.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
8	7	nfeq2 2780	. . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶
9	6, 8	nfan 1828	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)
10	9	nfoprab 6707	. 2 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
11	1, 10	nfcxfr 2762	1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751  {coprab 6651   ↦ cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  el2mpt2csbcl  7250  nfseq  12811  ptbasfi  21384  numclwlk1lem2  27230  sdclem1  33539  fmuldfeqlem1  39814  stoweidlem51  40268  vonicc  40899