Theorem nfof 7681

Description: Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypothesis

Ref	Expression
nfof.1	⊢ Ⅎ𝑥𝑅

Assertion

Ref	Expression
nfof	⊢ Ⅎ𝑥 ∘f 𝑅

Proof of Theorem nfof

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

Step	Hyp	Ref	Expression
1		df-of 7675	. 2 ⊢ ∘f 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
2		nfcv 2931	. . 3 ⊢ Ⅎ𝑥V
3		nfcv 2931	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
4		nfcv 2931	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
5		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
6		nfcv 2931	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
7	4, 5, 6	nfov 7441	. . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤))
8	3, 7	nfmpt 5213	. . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))
9	2, 2, 8	nfmpo 7493	. 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
10	1, 9	nfcxfr 2929	1 ⊢ Ⅎ𝑥 ∘f 𝑅

Syntax hints:  Ⅎwnfc 2916  Vcvv 3463   ∩ cin 3912   ↦ cmpt 5196  dom cdm 5662  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   ∘_f cof 7673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-iota 6493  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675

This theorem is referenced by: (None)