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Theorem nfwrecsOLD 8321
Description: Obsolete proof of nfwrecs 8320 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 8317 . 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 1909 . . . . . 6 𝑥 𝑓 Fn 𝑦
3 nfcv 2892 . . . . . . . 8 𝑥𝑦
4 nfwrecsOLD.2 . . . . . . . 8 𝑥𝐴
53, 4nfss 3964 . . . . . . 7 𝑥 𝑦𝐴
6 nfwrecsOLD.1 . . . . . . . . . 10 𝑥𝑅
7 nfcv 2892 . . . . . . . . . 10 𝑥𝑧
86, 4, 7nfpred 6305 . . . . . . . . 9 𝑥Pred(𝑅, 𝐴, 𝑧)
98, 3nfss 3964 . . . . . . . 8 𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
103, 9nfralw 3299 . . . . . . 7 𝑥𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
115, 10nfan 1894 . . . . . 6 𝑥(𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12 nfwrecsOLD.3 . . . . . . . . 9 𝑥𝐹
13 nfcv 2892 . . . . . . . . . 10 𝑥𝑓
1413, 8nfres 5981 . . . . . . . . 9 𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
1512, 14nffv 6902 . . . . . . . 8 𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
1615nfeq2 2910 . . . . . . 7 𝑥(𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
173, 16nfralw 3299 . . . . . 6 𝑥𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
182, 11, 17nf3an 1896 . . . . 5 𝑥(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
1918nfex 2312 . . . 4 𝑥𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
2019nfab 2898 . . 3 𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2120nfuni 4910 . 2 𝑥 {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦𝐴 ∧ ∀𝑧𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧𝑦 (𝑓𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
221, 21nfcxfr 2890 1 𝑥wrecs(𝑅, 𝐴, 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wa 394  w3a 1084   = wceq 1533  wex 1773  {cab 2702  wnfc 2875  wral 3051  wss 3939   cuni 4903  cres 5674  Predcpred 6299   Fn wfn 6538  cfv 6543  wrecscwrecs 8315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7419  df-2nd 7992  df-frecs 8285  df-wrecs 8316
This theorem is referenced by: (None)
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