Theorem nfmpo1 7490

Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Assertion

Ref	Expression
nfmpo1	⊢ Ⅎ𝑥(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Proof of Theorem nfmpo1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 7415	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab1 7471	. 2 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2923	1 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Ⅎwnfc 2910  {coprab 7411   ∈ cmpo 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-oprab 7414  df-mpo 7415

This theorem is referenced by:  ovmpos  7558  ov2gf  7559  ovmpodxf  7560  ovmpodf  7566  ovmpodv2  7568  xpcomco  9051  mapxpen  9127  pwfseqlem2  10639  pwfseqlem4a  10641  pwfseqlem4  10642  gsum2d2lem  20038  gsum2d2  20039  gsumcom2  20040  dprd2d2  20111  cnmpt21  23828  cnmpt2t  23830  cnmptcom  23835  cnmpt2k  23845  xkocnv  23971  numclwlk2lem2f1o  30730  finxpreclem2  38056  mnringmulrcld  44972  fmuldfeqlem1  46318  fmuldfeq  46319  ovmpordxf  49139