Theorem nfmpo 7502

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 7425	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		nfmpo.1	. . . . . 6 ⊢ Ⅎ𝑧𝐴
3	2	nfcri 2915	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
4		nfmpo.2	. . . . . 6 ⊢ Ⅎ𝑧𝐵
5	4	nfcri 2915	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
6	3, 5	nfan 1932	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7		nfmpo.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
8	7	nfeq2 2940	. . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶
9	6, 8	nfan 1932	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)
10	9	nfoprab 7484	. 2 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
11	1, 10	nfcxfr 2921	1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2908  {coprab 7421   ∈ cmpo 7422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-oprab 7424  df-mpo 7425

This theorem is used by:  nfof  7699  el2mpocsbcl  8096  nfseq  14154  ptbasfi  23900  nfseqs  28673  sdclem1  38677  fmuldfeqlem1  46593  stoweidlem51  47060  vonicc  47694