Theorem nfofr 7617

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 7611	. 2 ⊢ ∘r 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
2		nfcv 2894	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
3		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
4		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
5		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
6	3, 4, 5	nfbr 5136	. . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤)
7	2, 6	nfralw 3279	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)
8	7	nfopab 5158	. 2 ⊢ Ⅎ𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
9	1, 8	nfcxfr 2892	1 ⊢ Ⅎ𝑥 ∘r 𝑅

Syntax hints:  Ⅎwnfc 2879  ∀wral 3047   ∩ cin 3896   class class class wbr 5089  {copab 5151  dom cdm 5614  ‘cfv 6481   ∘_r cofr 7609

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-ofr 7611

This theorem is referenced by: (None)