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	⊢ Ⅎ𝑥𝑅
nfwrecsOLD.2	⊢ Ⅎ𝑥𝐴
nfwrecsOLD.3	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
nfwrecsOLD	⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

Proof of Theorem nfwrecsOLD

Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfwrecsOLD 8317	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦
3		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
4		nfwrecsOLD.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfss 3964	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
6		nfwrecsOLD.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅
7		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 4, 7	nfpred 6305	. . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧)
9	8, 3	nfss 3964	. . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
10	3, 9	nfralw 3299	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
11	5, 10	nfan 1894	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12		nfwrecsOLD.3	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
13		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓
14	13, 8	nfres 5981	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
15	12, 14	nffv 6902	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
16	15	nfeq2 2910	. . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
17	3, 16	nfralw 3299	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
18	2, 11, 17	nf3an 1896	. . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
19	18	nfex 2312	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
20	19	nfab 2898	. . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
21	20	nfuni 4910	. 2 ⊢ Ⅎ𝑥∪ {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
22	1, 21	nfcxfr 2890	1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

Syntax hints:   ∧ wa 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773  {cab 2702  Ⅎwnfc 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)