Theorem nfco 5804

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 5623	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
2		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑥𝑦
3		nfco.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
4		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 5136	. . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤
6		nfco.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
7		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑥𝑧
8	4, 6, 7	nfbr 5136	. . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧
9	5, 8	nfan 1900	. . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
10	9	nfex 2325	. . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
11	10	nfopab 5158	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
12	1, 11	nfcxfr 2892	1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 395  ∃wex 1780  Ⅎwnfc 2879   class class class wbr 5089  {copab 5151   ∘ ccom 5618

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-co 5623

This theorem is referenced by:  csbcog  6244  nffun  6504  nftpos  8191  nfwrecs  8244  cnmpt11  23578  cnmpt21  23586  poimirlem16  37686  poimirlem19  37689  choicefi  45307  cncficcgt0  45996  volioofmpt  46102  volicofmpt  46105  stoweidlem31  46139  stoweidlem59  46167