Theorem nfofr 6056

Description: Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypothesis

Ref	Expression
nfof.1	⊢ Ⅎ𝑥𝑅

Assertion

Ref	Expression
nfofr	⊢ Ⅎ𝑥 ∘𝑟 𝑅

Proof of Theorem nfofr

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ofr 6051	. 2 ⊢ ∘𝑟 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
2		nfcv 2308	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
3		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
4		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
5		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
6	3, 4, 5	nfbr 4028	. . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤)
7	2, 6	nfralxy 2504	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)
8	7	nfopab 4050	. 2 ⊢ Ⅎ𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
9	1, 8	nfcxfr 2305	1 ⊢ Ⅎ𝑥 ∘𝑟 𝑅

Syntax hints:  Ⅎwnfc 2295  ∀wral 2444   ∩ cin 3115   class class class wbr 3982  {copab 4042  dom cdm 4604  ‘cfv 5188   ∘_𝑟 cofr 6049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-ofr 6051

This theorem is referenced by: (None)