Theorem setrecseq 42231

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 6102	. . . . . . . . . 10 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑤) = (𝐺‘𝑤))
2	1	sseq1d 3595	. . . . . . . . 9 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑤) ⊆ 𝑧 ↔ (𝐺‘𝑤) ⊆ 𝑧))
3	2	imbi2d 329	. . . . . . . 8 ⊢ (𝐹 = 𝐺 → ((𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) ↔ (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)))
4	3	imbi2d 329	. . . . . . 7 ⊢ (𝐹 = 𝐺 → ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧))))
5	4	albidv 1836	. . . . . 6 ⊢ (𝐹 = 𝐺 → (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧))))
6	5	imbi1d 330	. . . . 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 1836	. . . 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 2728	. . 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 4382	. 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 42230	. 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
11		df-setrecs 42230	. 2 ⊢ setrecs(𝐺) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
12	9, 10, 11	3eqtr4g 2669	1 ⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))

Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475  {cab 2596   ⊆ wss 3540  ∪ cuni 4372  ‘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-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-rex 2902  df-in 3547  df-ss 3554  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-setrecs 42230

This theorem is referenced by: (None)