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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 42237 and setrec2v 42242
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 6945) 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 2751 | . . . . . 6 ⊢ Ⅎ𝑎𝑥 | |
3 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑎𝑢 | |
4 | 2, 1, 3 | nfbr 4629 | . . . . 5 ⊢ Ⅎ𝑎 𝑥𝐹𝑢 |
5 | 4 | nfeu 2474 | . . . 4 ⊢ Ⅎ𝑎∃!𝑢 𝑥𝐹𝑢 |
6 | 5 | nfab 2755 | . . 3 ⊢ Ⅎ𝑎{𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢} |
7 | 1, 6 | nfres 5319 | . 2 ⊢ Ⅎ𝑎(𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}) |
8 | setrec2.2 | . . 3 ⊢ 𝐵 = setrecs(𝐹) | |
9 | setrec2lem1 42239 | . . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) = (𝐹‘𝑤) | |
10 | 9 | sseq1i 3592 | . . . . . . . . . . 11 ⊢ (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧 ↔ (𝐹‘𝑤) ⊆ 𝑧) |
11 | 10 | imbi2i 325 | . . . . . . . . . 10 ⊢ ((𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧) ↔ (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
12 | 11 | imbi2i 325 | . . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))) |
13 | 12 | albii 1737 | . . . . . . . 8 ⊢ (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))) |
14 | 13 | imbi1i 338 | . . . . . . 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 1737 | . . . . . 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 2726 | . . . . 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 4381 | . . . 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 42230 | . . . 4 ⊢ setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
19 | df-setrecs 42230 | . . . 4 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
20 | 17, 18, 19 | 3eqtr4ri 2643 | . . 3 ⊢ setrecs(𝐹) = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) |
21 | 8, 20 | eqtri 2632 | . 2 ⊢ 𝐵 = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) |
22 | setrec2lem2 42240 | . 2 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}) | |
23 | setrec2.3 | . . 3 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) | |
24 | setrec2lem1 42239 | . . . . . 6 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) = (𝐹‘𝑎) | |
25 | 24 | sseq1i 3592 | . . . . 5 ⊢ (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶 ↔ (𝐹‘𝑎) ⊆ 𝐶) |
26 | 25 | imbi2i 325 | . . . 4 ⊢ ((𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ (𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) |
27 | 26 | albii 1737 | . . 3 ⊢ (∀𝑎(𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ ∀𝑎(𝑎 ⊆ 𝐶 → (𝐹‘𝑎) ⊆ 𝐶)) |
28 | 23, 27 | sylibr 223 | . 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶)) |
29 | 7, 21, 22, 28 | setrec2fun 42238 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 = wceq 1475 ∃!weu 2458 {cab 2596 Ⅎwnfc 2738 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 ↾ cres 5040 ‘cfv 5804 setrecscsetrecs 42229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-setrecs 42230 |
This theorem is referenced by: setrec2v 42242 |
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