Theorem nfof 7684

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 7678	. 2 ⊢ ∘f 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
2		nfcv 2922	. . 3 ⊢ Ⅎ𝑥V
3		nfcv 2922	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
4		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
5		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
6		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
7	4, 5, 6	nfov 7443	. . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤))
8	3, 7	nfmpt 5203	. . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))
9	2, 2, 8	nfmpo 7495	. 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
10	1, 9	nfcxfr 2920	1 ⊢ Ⅎ𝑥 ∘f 𝑅

This proof depends on syntax axioms:  Ⅎwnfc 2907  Vcvv 3450   ∩ cin 3898   ↦ cmpt 5186  dom cdm 5655  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415   ∘_f cof 7676

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-iota 6489  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678

This theorem is used by: (None)