Theorem nfmpo 7440

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

Hypotheses

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

Assertion

Ref	Expression
nfmpo	⊢ 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 nfmpo

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 7363	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		nfmpo.1	. . . . . 6 ⊢ Ⅎ𝑧𝐴
3	2	nfcri 2890	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
4		nfmpo.2	. . . . . 6 ⊢ Ⅎ𝑧𝐵
5	4	nfcri 2890	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
6	3, 5	nfan 1900	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7		nfmpo.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
8	7	nfeq2 2916	. . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶
9	6, 8	nfan 1900	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)
10	9	nfoprab 7422	. 2 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
11	1, 10	nfcxfr 2896	1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2883  {coprab 7359   ∈ cmpo 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-oprab 7362  df-mpo 7363

This theorem is referenced by:  nfof  7628  el2mpocsbcl  8027  nfseq  13936  ptbasfi  23527  nfseqs  28285  sdclem1  37946  fmuldfeqlem1  45849  stoweidlem51  46316  vonicc  46950