Theorem nfofr 6050

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 6045	. 2 ⊢ ∘𝑟 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
2		nfcv 2306	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
3		nfcv 2306	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
4		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
5		nfcv 2306	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
6	3, 4, 5	nfbr 4022	. . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤)
7	2, 6	nfralxy 2502	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)
8	7	nfopab 4044	. 2 ⊢ Ⅎ𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
9	1, 8	nfcxfr 2303	1 ⊢ Ⅎ𝑥 ∘𝑟 𝑅

Syntax hints:  Ⅎwnfc 2293  ∀wral 2442   ∩ cin 3110   class class class wbr 3976  {copab 4036  dom cdm 4598  ‘cfv 5182   ∘_𝑟 cofr 6043

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-ofr 6045

This theorem is referenced by: (None)