Theorem bnj126 34887

Description: Technical lemma for bnj150 34890. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj126.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj126.2	⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)
bnj126.3	⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
bnj126.4	⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}

Assertion

Ref	Expression
bnj126	⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

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

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝐴(𝑥,𝑦,𝑖)   𝑅(𝑥,𝑦,𝑖)   𝐹(𝑥,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓″(𝑥,𝑦,𝑓,𝑖,𝑛)

Proof of Theorem bnj126

Step	Hyp	Ref	Expression
1		bnj126.3	. 2 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
2		bnj126.2	. . 3 ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)
3	2	sbcbii 3846	. 2 ⊢ ([𝐹 / 𝑓]𝜓′ ↔ [𝐹 / 𝑓][1o / 𝑛]𝜓)
4		bnj126.1	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
5		bnj126.4	. . . 4 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
6	5	bnj95 34878	. . 3 ⊢ 𝐹 ∈ V
7	4, 6	bnj106 34882	. 2 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
8	1, 3, 7	3bitri 297	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∀wral 3061  [wsbc 3788  ∅c0 4333  {csn 4626  ⟨cop 4632  ∪ ciun 4991  suc csuc 6386  ‘cfv 6561  ωcom 7887  1_oc1o 8499   predc-bnj14 34702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908  df-iun 4993  df-br 5144  df-suc 6390  df-iota 6514  df-fv 6569  df-1o 8506

This theorem is referenced by:  bnj150  34890  bnj153  34894