Theorem nfmpt2 6622

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 6554	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		nfmpt2.1	. . . . . 6 ⊢ Ⅎ𝑧𝐴
3	2	nfcri 2745	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
4		nfmpt2.2	. . . . . 6 ⊢ Ⅎ𝑧𝐵
5	4	nfcri 2745	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
6	3, 5	nfan 1816	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7		nfmpt2.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
8	7	nfeq2 2766	. . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶
9	6, 8	nfan 1816	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)
10	9	nfoprab 6605	. 2 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
11	1, 10	nfcxfr 2749	1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  {coprab 6550   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  el2mpt2csbcl  7137  nfseq  12673  ptbasfi  21194  sdclem1  32709  fmuldfeqlem1  38649  stoweidlem51  38944  vonicc  39576