Theorem bnj126 31460

Description: Technical lemma for bnj150 31463. 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 3689	. 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 31451	. . 3 ⊢ 𝐹 ∈ V
7	4, 6	bnj106 31455	. 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 289	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157  ∀wral 3089  [wsbc 3633  ∅c0 4115  {csn 4368  ⟨cop 4374  ∪ ciun 4710  suc csuc 5943  ‘cfv 6101  ωcom 7299  1_𝑜c1o 7792   predc-bnj14 31274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-pw 4351  df-sn 4369  df-pr 4371  df-uni 4629  df-iun 4712  df-br 4844  df-suc 5947  df-iota 6064  df-fv 6109  df-1o 7799

This theorem is referenced by:  bnj150  31463  bnj153  31467