Theorem bnj229 34208

Description: Technical lemma for bnj517 34209. 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.)

Hypothesis

Ref	Expression
bnj229.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj229	⊢ ((𝑛 ∈ 𝑁 ∧ (suc 𝑚 = 𝑛 ∧ 𝑚 ∈ ω ∧ 𝜓)) → (𝐹‘𝑛) ⊆ 𝐴)

Distinct variable groups:   𝐴,𝑖,𝑚,𝑦   𝑖,𝐹,𝑚,𝑦   𝑖,𝑁,𝑚   𝑅,𝑖,𝑚

Allowed substitution hints:   𝜓(𝑦,𝑖,𝑚,𝑛)   𝐴(𝑛)   𝑅(𝑦,𝑛)   𝐹(𝑛)   𝑁(𝑦,𝑛)

Proof of Theorem bnj229

Step	Hyp	Ref	Expression
1		bnj213 34206	. . 3 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
2	1	bnj226 34058	. 2 ⊢ ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
3		bnj229.1	. . . . . . . 8 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4	3	bnj222 34207	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))
5	4	bnj228 34059	. . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝜓) → (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))
6	5	adantl 481	. . . . 5 ⊢ ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))
7		eleq1 2820	. . . . . . 7 ⊢ (suc 𝑚 = 𝑛 → (suc 𝑚 ∈ 𝑁 ↔ 𝑛 ∈ 𝑁))
8		fveqeq2 6900	. . . . . . 7 ⊢ (suc 𝑚 = 𝑛 → ((𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))
9	7, 8	imbi12d 344	. . . . . 6 ⊢ (suc 𝑚 = 𝑛 → ((suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛 ∈ 𝑁 → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))))
10	9	adantr 480	. . . . 5 ⊢ ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → ((suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛 ∈ 𝑁 → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))))
11	6, 10	mpbid 231	. . . 4 ⊢ ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (𝑛 ∈ 𝑁 → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))
12	11	3impb 1114	. . 3 ⊢ ((suc 𝑚 = 𝑛 ∧ 𝑚 ∈ ω ∧ 𝜓) → (𝑛 ∈ 𝑁 → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))
13	12	impcom 407	. 2 ⊢ ((𝑛 ∈ 𝑁 ∧ (suc 𝑚 = 𝑛 ∧ 𝑚 ∈ ω ∧ 𝜓)) → (𝐹‘𝑛) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))
14	2, 13	bnj1262 34134	1 ⊢ ((𝑛 ∈ 𝑁 ∧ (suc 𝑚 = 𝑛 ∧ 𝑚 ∈ ω ∧ 𝜓)) → (𝐹‘𝑛) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060   ⊆ wss 3948  ∪ ciun 4997  suc csuc 6366  ‘cfv 6543  ωcom 7859   predc-bnj14 34012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-suc 6370  df-iota 6495  df-fv 6551  df-bnj14 34013

This theorem is referenced by: (None)