Theorem nfco 5763

Description: Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)

Hypotheses

Ref	Expression
nfco.1	⊢ Ⅎ𝑥𝐴
nfco.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfco	⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)

Proof of Theorem nfco

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-co 5589	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
2		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑥𝑦
3		nfco.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
4		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 5117	. . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤
6		nfco.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
7		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑥𝑧
8	4, 6, 7	nfbr 5117	. . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧
9	5, 8	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
10	9	nfex 2322	. . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
11	10	nfopab 5139	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
12	1, 11	nfcxfr 2904	1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 395  ∃wex 1783  Ⅎwnfc 2886   class class class wbr 5070  {copab 5132   ∘ ccom 5584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-co 5589

This theorem is referenced by:  csbcog  6189  nffun  6441  nftpos  8048  nfwrecs  8103  cnmpt11  22722  cnmpt21  22730  poimirlem16  35720  poimirlem19  35723  choicefi  42629  cncficcgt0  43319  volioofmpt  43425  volicofmpt  43428  stoweidlem31  43462  stoweidlem59  43490