Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > nffrecs | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the founded recursion generator. (Contributed by Scott Fenton, 23-Dec-2021.) |
Ref | Expression |
---|---|
nffrecs.1 | ⊢ Ⅎ𝑥𝑅 |
nffrecs.2 | ⊢ Ⅎ𝑥𝐴 |
nffrecs.3 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
nffrecs | ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-frecs 33015 | . 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 1906 | . . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦 | |
3 | nfcv 2974 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
4 | nffrecs.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
5 | 3, 4 | nfss 3957 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴 |
6 | nffrecs.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅 | |
7 | nfcv 2974 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 4, 7 | nfpred 6146 | . . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) |
9 | 8, 3 | nfss 3957 | . . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦 |
10 | 3, 9 | nfralw 3222 | . . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦 |
11 | 5, 10 | nfan 1891 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) |
12 | nffrecs.3 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 | |
13 | nfcv 2974 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓 | |
14 | 13, 8 | nfres 5848 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)) |
15 | 7, 12, 14 | nfov 7175 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
16 | 15 | nfeq2 2992 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
17 | 3, 16 | nfralw 3222 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
18 | 2, 11, 17 | nf3an 1893 | . . . . 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 2334 | . . . 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 2981 | . . 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 4837 | . 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 2972 | 1 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∃wex 1771 {cab 2796 Ⅎwnfc 2958 ∀wral 3135 ⊆ wss 3933 ∪ cuni 4830 ↾ cres 5550 Predcpred 6140 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 frecscfrecs 33014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-iota 6307 df-fv 6356 df-ov 7148 df-frecs 33015 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |