Theorem nfof 7624

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 7618	. 2 ⊢ ∘f 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
2		nfcv 2895	. . 3 ⊢ Ⅎ𝑥V
3		nfcv 2895	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
4		nfcv 2895	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
5		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
6		nfcv 2895	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
7	4, 5, 6	nfov 7384	. . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤))
8	3, 7	nfmpt 5193	. . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))
9	2, 2, 8	nfmpo 7436	. 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
10	1, 9	nfcxfr 2893	1 ⊢ Ⅎ𝑥 ∘f 𝑅

Syntax hints:  Ⅎwnfc 2880  Vcvv 3437   ∩ cin 3897   ↦ cmpt 5176  dom cdm 5621  ‘cfv 6488  (class class class)co 7354   ∈ cmpo 7356   ∘_f cof 7616

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

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 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-iota 6444  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-of 7618

This theorem is referenced by: (None)