Theorem bnj518 35244

Description: Technical lemma for bnj852 35279. 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
bnj518.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj518.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj518.3	⊢ (𝜏 ↔ (𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛))

Assertion

Ref	Expression
bnj518	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝜏(𝑥,𝑦,𝑓,𝑖,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑛)   𝑅(𝑥,𝑓,𝑖,𝑛,𝑝)

Proof of Theorem bnj518

Step	Hyp	Ref	Expression
1		bnj518.3	. . . 4 ⊢ (𝜏 ↔ (𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛))
2		bnj334 35072	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛))
3	1, 2	bitri 278	. . 3 ⊢ (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛))
4		df-bnj17 35046	. . . 4 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛) ↔ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑝 ∈ 𝑛))
5		bnj518.1	. . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
6		bnj518.2	. . . . . 6 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
7	5, 6	bnj517 35243	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑝 ∈ 𝑛 (𝑓‘𝑝) ⊆ 𝐴)
8	7	r19.21bi 3264	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴)
9	4, 8	sylbi 220	. . 3 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴)
10	3, 9	sylbi 220	. 2 ⊢ (𝜏 → (𝑓‘𝑝) ⊆ 𝐴)
11		ssel 3939	. . . 4 ⊢ ((𝑓‘𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓‘𝑝) → 𝑦 ∈ 𝐴))
12		bnj93 35221	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V)
13	12	ex 417	. . . 4 ⊢ (𝑅 FrSe 𝐴 → (𝑦 ∈ 𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V))
14	11, 13	sylan9r 517	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓‘𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V))
15	14	ralrimiv 3163	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
16	10, 15	sylan2 604	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ∀wral 3086  Vcvv 3462   ⊆ wss 3913  ∅c0 4294  ∪ ciun 4961  suc csuc 6366  ‘cfv 6540  ωcom 7865   ∧ w-bnj17 35045   predc-bnj14 35047   FrSe w-bnj15 35051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-tr 5224  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fv 6548  df-om 7866  df-bnj17 35046  df-bnj14 35048  df-bnj13 35050  df-bnj15 35052

This theorem is referenced by:  bnj535  35248  bnj546  35254