Theorem nffrecs 8276

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 8274	. 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 1944	. . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦
3		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
4		nffrecs.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfss 3930	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
6		nffrecs.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅
7		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 4, 7	nfpred 6307	. . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧)
9	8, 3	nfss 3930	. . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
10	3, 9	nfralw 3312	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
11	5, 10	nfan 1929	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12		nffrecs.3	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
13		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓
14	13, 8	nfres 5980	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
15	7, 12, 14	nfov 7440	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
16	15	nfeq2 2942	. . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
17	3, 16	nfralw 3312	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
18	2, 11, 17	nf3an 1931	. . . . 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 2357	. . . 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 2931	. . 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 4879	. 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 2923	1 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹)

Syntax hints:   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809  {cab 2741  Ⅎwnfc 2910  ∀wral 3079   ⊆ wss 3905  ∪ cuni 4872   ↾ cres 5663  Predcpred 6301   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  frecscfrecs 8273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-iota 6492  df-fv 6544  df-ov 7413  df-frecs 8274

This theorem is referenced by:  nfwrecs  8307