Theorem nfco 5852

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 5671	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
2		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑥𝑦
3		nfco.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
4		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 5162	. . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤
6		nfco.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
7		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑥𝑧
8	4, 6, 7	nfbr 5162	. . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧
9	5, 8	nfan 1926	. . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
10	9	nfex 2363	. . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
11	10	nfopab 5184	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
12	1, 11	nfcxfr 2929	1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 400  ∃wex 1806  Ⅎwnfc 2916   class class class wbr 5113  {copab 5177   ∘ ccom 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-co 5671

This theorem is referenced by:  csbcog  6299  nffun  6560  nftpos  8257  nfwrecs  8311  cnmpt11  23789  cnmpt21  23797  poimirlem16  38175  poimirlem19  38178  choicefi  45809  cncficcgt0  46494  volioofmpt  46600  volicofmpt  46603  stoweidlem31  46637  stoweidlem59  46665