Theorem setrecseq 46391

Description: Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)

Assertion

Ref	Expression
setrecseq	⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))

Proof of Theorem setrecseq

Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6773	. . . . . . . . . 10 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑤) = (𝐺‘𝑤))
2	1	sseq1d 3952	. . . . . . . . 9 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑤) ⊆ 𝑧 ↔ (𝐺‘𝑤) ⊆ 𝑧))
3	2	imbi2d 341	. . . . . . . 8 ⊢ (𝐹 = 𝐺 → ((𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) ↔ (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)))
4	3	imbi2d 341	. . . . . . 7 ⊢ (𝐹 = 𝐺 → ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧))))
5	4	albidv 1923	. . . . . 6 ⊢ (𝐹 = 𝐺 → (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧))))
6	5	imbi1d 342	. . . . 5 ⊢ (𝐹 = 𝐺 → ((∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)))
7	6	albidv 1923	. . . 4 ⊢ (𝐹 = 𝐺 → (∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)))
8	7	abbidv 2807	. . 3 ⊢ (𝐹 = 𝐺 → {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
9	8	unieqd 4853	. 2 ⊢ (𝐹 = 𝐺 → ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})
10		df-setrecs 46390	. 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
11		df-setrecs 46390	. 2 ⊢ setrecs(𝐺) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
12	9, 10, 11	3eqtr4g 2803	1 ⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))

Syntax hints:   → wi 4  ∀wal 1537   = wceq 1539  {cab 2715   ⊆ wss 3887  ∪ cuni 4839  ‘cfv 6433  setrecscsetrecs 46389

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-setrecs 46390

This theorem is referenced by: (None)