Theorem nfco 5856

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 5676	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
2		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑥𝑦
3		nfco.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
4		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 5186	. . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤
6		nfco.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
7		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑥𝑧
8	4, 6, 7	nfbr 5186	. . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧
9	5, 8	nfan 1894	. . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
10	9	nfex 2309	. . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
11	10	nfopab 5208	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
12	1, 11	nfcxfr 2893	1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 395  ∃wex 1773  Ⅎwnfc 2875   class class class wbr 5139  {copab 5201   ∘ ccom 5671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-co 5676

This theorem is referenced by:  csbcog  6287  nffun  6562  nftpos  8242  nfwrecs  8297  cnmpt11  23491  cnmpt21  23499  poimirlem16  36998  poimirlem19  37001  choicefi  44409  cncficcgt0  45114  volioofmpt  45220  volicofmpt  45223  stoweidlem31  45257  stoweidlem59  45285