Theorem nfof 7659

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 7653	. 2 ⊢ ∘f 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
2		nfcv 2891	. . 3 ⊢ Ⅎ𝑥V
3		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
4		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
5		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
6		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
7	4, 5, 6	nfov 7417	. . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤))
8	3, 7	nfmpt 5205	. . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))
9	2, 2, 8	nfmpo 7471	. 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
10	1, 9	nfcxfr 2889	1 ⊢ Ⅎ𝑥 ∘f 𝑅

Syntax hints:  Ⅎwnfc 2876  Vcvv 3447   ∩ cin 3913   ↦ cmpt 5188  dom cdm 5638  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389   ∘_f cof 7651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-iota 6464  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653

This theorem is referenced by: (None)