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Theorem nfwrecsOLD 8249
Description: Obsolete proof of nfwrecs 8248 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 9-Jun-2018.)
Hypotheses
Ref Expression
nfwrecsOLD.1 𝑥𝑅
nfwrecsOLD.2 𝑥𝐴
nfwrecsOLD.3 𝑥𝐹
Assertion
Ref Expression
nfwrecsOLD 𝑥wrecs(𝑅, 𝐴, 𝐹)

Proof of Theorem nfwrecsOLD
Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfwrecsOLD 8245 . 2 wrecs(𝑅, 𝐴, 𝐹) = {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2 nfv 1918 . . . . . 6 𝑥 𝑓 Fn 𝑦
3 nfcv 2904 . . . . . . . 8 𝑥𝑦
4 nfwrecsOLD.2 . . . . . . . 8 𝑥𝐴
53, 4nfss 3937 . . . . . . 7 𝑥 𝑦𝐴
6 nfwrecsOLD.1 . . . . . . . . . 10 𝑥𝑅
7 nfcv 2904 . . . . . . . . . 10 𝑥𝑧
86, 4, 7nfpred 6259 . . . . . . . . 9 𝑥Pred(𝑅, 𝐴, 𝑧)
98, 3nfss 3937 . . . . . . . 8 𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
103, 9nfralw 3293 . . . . . . 7 𝑥𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
115, 10nfan 1903 . . . . . 6 𝑥(𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12 nfwrecsOLD.3 . . . . . . . . 9 𝑥𝐹
13 nfcv 2904 . . . . . . . . . 10 𝑥𝑓
1413, 8nfres 5940 . . . . . . . . 9 𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
1512, 14nffv 6853 . . . . . . . 8 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
1615nfeq2 2921 . . . . . . 7 𝑥(𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
173, 16nfralw 3293 . . . . . 6 𝑥𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
182, 11, 17nf3an 1905 . . . . 5 𝑥(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
1918nfex 2318 . . . 4 𝑥𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
2019nfab 2910 . . 3 𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2120nfuni 4873 . 2 𝑥 {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
221, 21nfcxfr 2902 1 𝑥wrecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wa 397  w3a 1088   = wceq 1542  wex 1782  {cab 2710  wnfc 2884  wral 3061  wss 3911   cuni 4866  cres 5636  Predcpred 6253   Fn wfn 6492  cfv 6497  wrecscwrecs 8243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244
This theorem is referenced by: (None)
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