Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec1lem1 | Structured version Visualization version GIF version |
Description: Lemma for setrec1 46068. This is a utility theorem showing the
equivalence
of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof
uses
elabg 3585 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 | sseq2 3927 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑤 ⊆ 𝑦 ↔ 𝑤 ⊆ 𝑋)) | |
3 | 2 | imbi1d 345 | . . . . . 6 ⊢ (𝑦 = 𝑋 → ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))) |
4 | 3 | albidv 1928 | . . . . 5 ⊢ (𝑦 = 𝑋 → (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))) |
5 | sseq1 3926 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑧)) | |
6 | 4, 5 | imbi12d 348 | . . . 4 ⊢ (𝑦 = 𝑋 → ((∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
7 | 6 | albidv 1928 | . . 3 ⊢ (𝑦 = 𝑋 → (∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
8 | setrec1lem1.1 | . . 3 ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
9 | 7, 8 | elab2g 3589 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))) |
10 | 1, 9 | 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 209 ∀wal 1541 = wceq 1543 ∈ wcel 2110 {cab 2714 ⊆ wss 3866 ‘cfv 6380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-in 3873 df-ss 3883 |
This theorem is referenced by: setrec1lem2 46065 setrec1lem4 46067 setrec2fun 46069 |
Copyright terms: Public domain | W3C validator |