Theorem nffrecs 8227

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 8225	. 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 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦
3		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
4		nffrecs.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
5	3, 4	nfss 3915	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴
6		nffrecs.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅
7		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 4, 7	nfpred 6265	. . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧)
9	8, 3	nfss 3915	. . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
10	3, 9	nfralw 3285	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦
11	5, 10	nfan 1901	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦)
12		nffrecs.3	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
13		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓
14	13, 8	nfres 5941	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))
15	7, 12, 14	nfov 7391	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
16	15	nfeq2 2917	. . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
17	3, 16	nfralw 3285	. . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))
18	2, 11, 17	nf3an 1903	. . . . 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 2330	. . . 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 2905	. . 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 4858	. 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 2897	1 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹)

Syntax hints:   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781  {cab 2715  Ⅎwnfc 2884  ∀wral 3052   ⊆ wss 3890  ∪ cuni 4851   ↾ cres 5627  Predcpred 6259   Fn wfn 6488  ‘cfv 6493  (class class class)co 7361  frecscfrecs 8224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-iota 6449  df-fv 6501  df-ov 7364  df-frecs 8225

This theorem is referenced by:  nfwrecs  8258