Theorem nffrecs 8099

Description: Bound-variable hypothesis builder for the well-founded recursion generator. (Contributed by Scott Fenton, 23-Dec-2021.)

Hypotheses

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

Assertion

Ref	Expression
nffrecs	⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹)

Proof of Theorem nffrecs

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

Step	Hyp	Ref	Expression
1		df-frecs 8097	. 2 ⊢ frecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
2		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦
3		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
4		nffrecs.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfss 3913	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
6		nffrecs.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅
7		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 4, 7	nfpred 6207	. . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧)
9	8, 3	nfss 3913	. . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
10	3, 9	nfralw 3151	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
11	5, 10	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12		nffrecs.3	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
13		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓
14	13, 8	nfres 5893	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
15	7, 12, 14	nfov 7305	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
16	15	nfeq2 2924	. . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
17	3, 16	nfralw 3151	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
18	2, 11, 17	nf3an 1904	. . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
19	18	nfex 2318	. . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))
20	19	nfab 2913	. . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
21	20	nfuni 4846	. 2 ⊢ Ⅎ𝑥∪ {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))}
22	1, 21	nfcxfr 2905	1 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹)

Syntax hints:   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782  {cab 2715  Ⅎwnfc 2887  ∀wral 3064   ⊆ wss 3887  ∪ cuni 4839   ↾ cres 5591  Predcpred 6201   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  frecscfrecs 8096

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-iota 6391  df-fv 6441  df-ov 7278  df-frecs 8097

This theorem is referenced by:  nfwrecs  8132