Theorem nfof 5877

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 5872	. 2 ⊢ ∘𝑓 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
2		nfcv 2229	. . 3 ⊢ Ⅎ𝑥V
3		nfcv 2229	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
4		nfcv 2229	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
5		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
6		nfcv 2229	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
7	4, 5, 6	nfov 5695	. . . 4 ⊢ Ⅎ𝑥((𝑢‘𝑤)𝑅(𝑣‘𝑤))
8	3, 7	nfmpt 3938	. . 3 ⊢ Ⅎ𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤)))
9	2, 2, 8	nfmpt2 5733	. 2 ⊢ Ⅎ𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢‘𝑤)𝑅(𝑣‘𝑤))))
10	1, 9	nfcxfr 2226	1 ⊢ Ⅎ𝑥 ∘𝑓 𝑅

Syntax hints:  Ⅎwnfc 2216  Vcvv 2622   ∩ cin 3001   ↦ cmpt 3907  dom cdm 4454  ‘cfv 5030  (class class class)co 5668   ↦ cmpt2 5670   ∘_𝑓 cof 5870

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2624  df-un 3006  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-iota 4995  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-of 5872

This theorem is referenced by: (None)