Theorem setrecseq 50619

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 6881	. . . . . . . . . 10 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑤) = (𝐺‘𝑤))
2	1	sseq1d 3965	. . . . . . . . 9 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑤) ⊆ 𝑧 ↔ (𝐺‘𝑤) ⊆ 𝑧))
3	2	imbi2d 343	. . . . . . . 8 ⊢ (𝐹 = 𝐺 → ((𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) ↔ (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)))
4	3	imbi2d 343	. . . . . . 7 ⊢ (𝐹 = 𝐺 → ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧))))
5	4	albidv 1953	. . . . . 6 ⊢ (𝐹 = 𝐺 → (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧))))
6	5	imbi1d 344	. . . . 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 1953	. . . 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 2828	. . 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 4883	. 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 50618	. 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
11		df-setrecs 50618	. 2 ⊢ setrecs(𝐺) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐺‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
12	9, 10, 11	3eqtr4g 2822	1 ⊢ (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570  {cab 2740   ⊆ wss 3902  ∪ cuni 4870  ‘cfv 6537  setrecscsetrecs 50617

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-ss 3919  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-setrecs 50618

This theorem is used by: (None)