Theorem nfofr 5988

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 5983	. 2 ⊢ ∘𝑟 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
2		nfcv 2281	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
3		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
4		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
5		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
6	3, 4, 5	nfbr 3974	. . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤)
7	2, 6	nfralxy 2471	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)
8	7	nfopab 3996	. 2 ⊢ Ⅎ𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
9	1, 8	nfcxfr 2278	1 ⊢ Ⅎ𝑥 ∘𝑟 𝑅

Syntax hints:  Ⅎwnfc 2268  ∀wral 2416   ∩ cin 3070   class class class wbr 3929  {copab 3988  dom cdm 4539  ‘cfv 5123   ∘_𝑟 cofr 5981

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-ofr 5983

This theorem is referenced by: (None)