Theorem bnj92 34626

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 3834	. 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 34504	. 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 3825	. . . . 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 3845	. . . . . 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 34502	. . . . 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 349	. . . 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 3082	. 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 296	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 1533   ∈ wcel 2098  ∀wral 3050  Vcvv 3461  [wsbc 3773  ∪ ciun 4997  suc csuc 6373  ‘cfv 6549  ωcom 7871   predc-bnj14 34452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-v 3463  df-sbc 3774

This theorem is referenced by:  bnj106  34632  bnj153  34644