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Theorem nfof 7703
Description: Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypothesis
Ref Expression
nfof.1 𝑥𝑅
Assertion
Ref Expression
nfof 𝑥f 𝑅

Proof of Theorem nfof
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-of 7697 . 2 f 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤))))
2 nfcv 2903 . . 3 𝑥V
3 nfcv 2903 . . . 4 𝑥(dom 𝑢 ∩ dom 𝑣)
4 nfcv 2903 . . . . 5 𝑥(𝑢𝑤)
5 nfof.1 . . . . 5 𝑥𝑅
6 nfcv 2903 . . . . 5 𝑥(𝑣𝑤)
74, 5, 6nfov 7461 . . . 4 𝑥((𝑢𝑤)𝑅(𝑣𝑤))
83, 7nfmpt 5255 . . 3 𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤)))
92, 2, 8nfmpo 7515 . 2 𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤))))
101, 9nfcxfr 2901 1 𝑥f 𝑅
Colors of variables: wff setvar class
Syntax hints:  wnfc 2888  Vcvv 3478  cin 3962  cmpt 5231  dom cdm 5689  cfv 6563  (class class class)co 7431  cmpo 7433  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by: (None)
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