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Mirrors > Home > MPE Home > Th. List > nffrecs | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the well-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 8216 | . 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 1918 | . . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦 | |
3 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
4 | nffrecs.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
5 | 3, 4 | nfss 3940 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴 |
6 | nffrecs.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅 | |
7 | nfcv 2904 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 4, 7 | nfpred 6262 | . . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) |
9 | 8, 3 | nfss 3940 | . . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦 |
10 | 3, 9 | nfralw 3293 | . . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦 |
11 | 5, 10 | nfan 1903 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) |
12 | nffrecs.3 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 | |
13 | nfcv 2904 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓 | |
14 | 13, 8 | nfres 5943 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)) |
15 | 7, 12, 14 | nfov 7391 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
16 | 15 | nfeq2 2921 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
17 | 3, 16 | nfralw 3293 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝑧𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
18 | 2, 11, 17 | nf3an 1905 | . . . . 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 2910 | . . 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 4876 | . 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 2902 | 1 ⊢ Ⅎ𝑥frecs(𝑅, 𝐴, 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 {cab 2710 Ⅎwnfc 2884 ∀wral 3061 ⊆ wss 3914 ∪ cuni 4869 ↾ cres 5639 Predcpred 6256 Fn wfn 6495 ‘cfv 6500 (class class class)co 7361 frecscfrecs 8215 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-xp 5643 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-iota 6452 df-fv 6508 df-ov 7364 df-frecs 8216 |
This theorem is referenced by: nfwrecs 8251 |
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