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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec2 | Structured version Visualization version GIF version |
Description: This is the second of two
fundamental theorems about set recursion from
which all other facts will be derived. It states that the class
setrecs(𝐹) is a subclass of all classes 𝐶 that
are closed
under 𝐹. Taken together, theorems setrec1 42438 and setrec2v 42443
uniquely determine setrecs(𝐹) to be the minimal class closed
under 𝐹.
We express this by saying that if 𝐹 respects the ⊆ relation and 𝐶 is closed under 𝐹, then 𝐵 ⊆ 𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7053) to the other class. (Contributed by Emmett Weisz, 2-Sep-2021.) |
Ref | Expression |
---|---|
setrec2.1 | ⊢ Ⅎ𝑎𝐹 |
setrec2.2 | ⊢ 𝐵 = setrecs(𝐹) |
setrec2.3 | ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) |
Ref | Expression |
---|---|
setrec2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setrec2.1 | . . 3 ⊢ Ⅎ𝑎𝐹 | |
2 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑎𝑥 | |
3 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑎𝑢 | |
4 | 2, 1, 3 | nfbr 4699 | . . . . 5 ⊢ Ⅎ𝑎 𝑥𝐹𝑢 |
5 | 4 | nfeu 2486 | . . . 4 ⊢ Ⅎ𝑎∃!𝑢 𝑥𝐹𝑢 |
6 | 5 | nfab 2769 | . . 3 ⊢ Ⅎ𝑎{𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢} |
7 | 1, 6 | nfres 5398 | . 2 ⊢ Ⅎ𝑎(𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}) |
8 | setrec2.2 | . . 3 ⊢ 𝐵 = setrecs(𝐹) | |
9 | setrec2lem1 42440 | . . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) = (𝐹‘𝑤) | |
10 | 9 | sseq1i 3629 | . . . . . . . . . . 11 ⊢ (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧 ↔ (𝐹‘𝑤) ⊆ 𝑧) |
11 | 10 | imbi2i 326 | . . . . . . . . . 10 ⊢ ((𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧) ↔ (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
12 | 11 | imbi2i 326 | . . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))) |
13 | 12 | albii 1747 | . . . . . . . 8 ⊢ (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))) |
14 | 13 | imbi1i 339 | . . . . . . 7 ⊢ ((∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)) |
15 | 14 | albii 1747 | . . . . . 6 ⊢ (∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)) |
16 | 15 | abbii 2739 | . . . . 5 ⊢ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
17 | 16 | unieqi 4445 | . . . 4 ⊢ ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
18 | df-setrecs 42431 | . . . 4 ⊢ setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
19 | df-setrecs 42431 | . . . 4 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
20 | 17, 18, 19 | 3eqtr4ri 2655 | . . 3 ⊢ setrecs(𝐹) = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) |
21 | 8, 20 | eqtri 2644 | . 2 ⊢ 𝐵 = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) |
22 | setrec2lem2 42441 | . 2 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}) | |
23 | setrec2.3 | . . 3 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) | |
24 | setrec2lem1 42440 | . . . . . 6 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) = (𝐹‘𝑎) | |
25 | 24 | sseq1i 3629 | . . . . 5 ⊢ (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶 ↔ (𝐹‘𝑎) ⊆ 𝐶) |
26 | 25 | imbi2i 326 | . . . 4 ⊢ ((𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ (𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) |
27 | 26 | albii 1747 | . . 3 ⊢ (∀𝑎(𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) |
28 | 23, 27 | sylibr 224 | . 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶)) |
29 | 7, 21, 22, 28 | setrec2fun 42439 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 = wceq 1483 ∃!weu 2470 {cab 2608 Ⅎwnfc 2751 ⊆ wss 3574 ∪ cuni 4436 class class class wbr 4653 ↾ cres 5116 ‘cfv 5888 setrecscsetrecs 42430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-setrecs 42431 |
This theorem is referenced by: setrec2v 42443 |
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