Theorem bnj953 32206

Description: Technical lemma for bnj69 32277. 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
bnj953.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj953.2	⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))

Assertion

Ref	Expression
bnj953	⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))

Proof of Theorem bnj953

Step	Hyp	Ref	Expression
1		bnj312 31977	. . 3 ⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓))
2		bnj251 31967	. . 3 ⊢ (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺‘𝑖) = (𝑓‘𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓))))
3	1, 2	bitri 277	. 2 ⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺‘𝑖) = (𝑓‘𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓))))
4		bnj953.1	. . . . . 6 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
5	4	bnj115 31990	. . . . 5 ⊢ (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
6		sp 2177	. . . . . 6 ⊢ (∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) → ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
7	6	impcom 410	. . . . 5 ⊢ (((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
8	5, 7	sylan2b 595	. . . 4 ⊢ (((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
9		bnj953.2	. . . . 5 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))
10	9	bnj956 32043	. . . 4 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
11		eqtr3 2843	. . . 4 ⊢ (((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ∧ ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
12	8, 10, 11	syl2anr 598	. . 3 ⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
13		eqtr 2841	. . 3 ⊢ (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
14	12, 13	sylan2 594	. 2 ⊢ (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺‘𝑖) = (𝑓‘𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓))) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
15	3, 14	sylbi 219	1 ⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∪ ciun 4912  suc csuc 6188  ‘cfv 6350  ωcom 7574   ∧ w-bnj17 31951   predc-bnj14 31953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ral 3143  df-rex 3144  df-iun 4914  df-bnj17 31952

This theorem is referenced by:  bnj967  32212