Theorem nfco 5736

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 5564	. 2 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
2		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝑦
3		nfco.2	. . . . . 6 ⊢ Ⅎ𝑥𝐵
4		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 5113	. . . . 5 ⊢ Ⅎ𝑥 𝑦𝐵𝑤
6		nfco.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
7		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝑧
8	4, 6, 7	nfbr 5113	. . . . 5 ⊢ Ⅎ𝑥 𝑤𝐴𝑧
9	5, 8	nfan 1900	. . . 4 ⊢ Ⅎ𝑥(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
10	9	nfex 2343	. . 3 ⊢ Ⅎ𝑥∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)
11	10	nfopab 5134	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧)}
12	1, 11	nfcxfr 2975	1 ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 398  ∃wex 1780  Ⅎwnfc 2961   class class class wbr 5066  {copab 5128   ∘ ccom 5559

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-co 5564

This theorem is referenced by:  nffun  6378  nftpos  7927  cnmpt11  22271  cnmpt21  22279  poimirlem16  34923  poimirlem19  34926  csbcog  40043  choicefi  41512  cncficcgt0  42220  volioofmpt  42328  volicofmpt  42331  stoweidlem31  42365  stoweidlem59  42393