Theorem nfwrecs 7296

Description: Bound-variable hypothesis builder for the well-founded recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.)

Hypotheses

Ref	Expression
nfwrecs.1	⊢ Ⅎ𝑥𝑅
nfwrecs.2	⊢ Ⅎ𝑥𝐴
nfwrecs.3	⊢ Ⅎ𝑥𝐹

Assertion

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

Proof of Theorem nfwrecs

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

Step	Hyp	Ref	Expression
1		df-wrecs 7294	. 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 1830	. . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦
3		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
4		nfwrecs.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfss 3561	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
6		nfwrecs.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅
7		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 4, 7	nfpred 5602	. . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧)
9	8, 3	nfss 3561	. . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
10	3, 9	nfral 2929	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
11	5, 10	nfan 1816	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12		nfwrecs.3	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
13		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓
14	13, 8	nfres 5319	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
15	12, 14	nffv 6110	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
16	15	nfeq2 2766	. . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
17	3, 16	nfral 2929	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
18	2, 11, 17	nf3an 1819	. . . . 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 2140	. . . 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 2755	. . 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 4378	. 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 2749	1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

Syntax hints:   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695  {cab 2596  Ⅎwnfc 2738  ∀wral 2896   ⊆ wss 3540  ∪ cuni 4372   ↾ cres 5040  Predcpred 5596   Fn wfn 5799  ‘cfv 5804  wrecscwrecs 7293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-wrecs 7294

This theorem is referenced by:  nfrecs  7358