Theorem bnj92 32842

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj92.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj92.2	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
bnj92	⊢ ([𝑍 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑍 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑖,𝑍   𝑓,𝑛   𝑖,𝑛   𝑦,𝑛

Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑓,𝑖)   𝑅(𝑦,𝑓,𝑖)   𝑍(𝑦,𝑓,𝑛)

Proof of Theorem bnj92

Step	Hyp	Ref	Expression
1		bnj92.1	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
2	1	sbcbii 3776	. 2 ⊢ ([𝑍 / 𝑛]𝜓 ↔ [𝑍 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj92.2	. . 3 ⊢ 𝑍 ∈ V
4	3	bnj538 32720	. 2 ⊢ ([𝑍 / 𝑛]∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω [𝑍 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
5		sbcimg 3767	. . . . 5 ⊢ (𝑍 ∈ V → ([𝑍 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝑍 / 𝑛]suc 𝑖 ∈ 𝑛 → [𝑍 / 𝑛](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
6	3, 5	ax-mp 5	. . . 4 ⊢ ([𝑍 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝑍 / 𝑛]suc 𝑖 ∈ 𝑛 → [𝑍 / 𝑛](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
7		sbcel2gv 3788	. . . . . 6 ⊢ (𝑍 ∈ V → ([𝑍 / 𝑛]suc 𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑍))
8	3, 7	ax-mp 5	. . . . 5 ⊢ ([𝑍 / 𝑛]suc 𝑖 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑍)
9	3	bnj525 32718	. . . . 5 ⊢ ([𝑍 / 𝑛](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
10	8, 9	imbi12i 351	. . . 4 ⊢ (([𝑍 / 𝑛]suc 𝑖 ∈ 𝑛 → [𝑍 / 𝑛](𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑍 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
11	6, 10	bitri 274	. . 3 ⊢ ([𝑍 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑍 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
12	11	ralbii 3092	. 2 ⊢ (∀𝑖 ∈ ω [𝑍 / 𝑛](suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑍 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
13	2, 4, 12	3bitri 297	1 ⊢ ([𝑍 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑍 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  [wsbc 3716  ∪ ciun 4924  suc csuc 6268  ‘cfv 6433  ωcom 7712   predc-bnj14 32667

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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-v 3434  df-sbc 3717

This theorem is referenced by:  bnj106  32848  bnj153  32860