Theorem bnj126 35003

Description: Technical lemma for bnj150 35006. 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 3781	. 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 34994	. . 3 ⊢ 𝐹 ∈ V
7	4, 6	bnj106 34998	. 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 1542   ∈ wcel 2114  ∀wral 3049  [wsbc 3725  ∅c0 4263  {csn 4557  ⟨cop 4563  ∪ ciun 4923  suc csuc 6314  ‘cfv 6487  ωcom 7806  1_oc1o 8387   predc-bnj14 34819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3050  df-rex 3060  df-v 3429  df-sbc 3726  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4264  df-pw 4533  df-sn 4558  df-pr 4560  df-uni 4841  df-iun 4925  df-br 5075  df-suc 6318  df-iota 6443  df-fv 6495  df-1o 8394

This theorem is referenced by:  bnj150  35006  bnj153  35010