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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec1lem1 | Structured version Visualization version GIF version |
Description: Lemma for setrec1 42237. This is a utility theorem showing the
equivalence
of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof
uses
elabg 3320 and equivalence theorems.
Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦 ∈ 𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
setrec1lem1.1 | ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
setrec1lem1.2 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
setrec1lem1 | ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setrec1lem1.2 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | setrec1lem1.1 | . . . 4 ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
3 | 2 | eleq2i 2680 | . . 3 ⊢ (𝑋 ∈ 𝑌 ↔ 𝑋 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}) |
4 | sseq2 3590 | . . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑤 ⊆ 𝑦 ↔ 𝑤 ⊆ 𝑋)) | |
5 | 4 | imbi1d 330 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))) |
6 | 5 | albidv 1836 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))) |
7 | sseq1 3589 | . . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑧)) | |
8 | 6, 7 | imbi12d 333 | . . . . 5 ⊢ (𝑦 = 𝑋 → ((∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
9 | 8 | albidv 1836 | . . . 4 ⊢ (𝑦 = 𝑋 → (∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
10 | 9 | elabg 3320 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
11 | 3, 10 | syl5bb 271 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
12 | 1, 11 | syl 17 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 {cab 2596 ⊆ wss 3540 ‘cfv 5804 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 |
This theorem is referenced by: setrec1lem2 42234 setrec1lem4 42236 setrec2fun 42238 |
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