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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 33118 | . 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 1915 | . . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦 | |
3 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
4 | nffrecs.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
5 | 3, 4 | nfss 3960 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴 |
6 | nffrecs.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅 | |
7 | nfcv 2977 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 4, 7 | nfpred 6153 | . . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) |
9 | 8, 3 | nfss 3960 | . . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦 |
10 | 3, 9 | nfralw 3225 | . . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦 |
11 | 5, 10 | nfan 1900 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) |
12 | nffrecs.3 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 | |
13 | nfcv 2977 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓 | |
14 | 13, 8 | nfres 5855 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)) |
15 | 7, 12, 14 | nfov 7186 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
16 | 15 | nfeq2 2995 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
17 | 3, 16 | nfralw 3225 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
18 | 2, 11, 17 | nf3an 1902 | . . . . 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 2343 | . . . 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 2984 | . . 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 4845 | . 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 2975 | 1 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 {cab 2799 Ⅎwnfc 2961 ∀wral 3138 ⊆ wss 3936 ∪ cuni 4838 ↾ cres 5557 Predcpred 6147 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 frecscfrecs 33117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-iota 6314 df-fv 6363 df-ov 7159 df-frecs 33118 |
This theorem is referenced by: (None) |
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