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Theorem nfof 7680
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 7674 . 2 f 𝑅 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤))))
2 nfcv 2902 . . 3 𝑥V
3 nfcv 2902 . . . 4 𝑥(dom 𝑢 ∩ dom 𝑣)
4 nfcv 2902 . . . . 5 𝑥(𝑢𝑤)
5 nfof.1 . . . . 5 𝑥𝑅
6 nfcv 2902 . . . . 5 𝑥(𝑣𝑤)
74, 5, 6nfov 7442 . . . 4 𝑥((𝑢𝑤)𝑅(𝑣𝑤))
83, 7nfmpt 5255 . . 3 𝑥(𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤)))
92, 2, 8nfmpo 7494 . 2 𝑥(𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑤 ∈ (dom 𝑢 ∩ dom 𝑣) ↦ ((𝑢𝑤)𝑅(𝑣𝑤))))
101, 9nfcxfr 2900 1 𝑥f 𝑅
Colors of variables: wff setvar class
Syntax hints:  wnfc 2882  Vcvv 3473  cin 3947  cmpt 5231  dom cdm 5676  cfv 6543  (class class class)co 7412  cmpo 7414  f cof 7672
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6495  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674
This theorem is referenced by: (None)
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