Theorem bnj518 34876

Description: Technical lemma for bnj852 34911. 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 34703	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛))
3	1, 2	bitri 275	. . 3 ⊢ (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛))
4		df-bnj17 34677	. . . 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 34875	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑝 ∈ 𝑛 (𝑓‘𝑝) ⊆ 𝐴)
8	7	r19.21bi 3229	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴)
9	4, 8	sylbi 217	. . 3 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴)
10	3, 9	sylbi 217	. 2 ⊢ (𝜏 → (𝑓‘𝑝) ⊆ 𝐴)
11		ssel 3940	. . . 4 ⊢ ((𝑓‘𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓‘𝑝) → 𝑦 ∈ 𝐴))
12		bnj93 34853	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V)
13	12	ex 412	. . . 4 ⊢ (𝑅 FrSe 𝐴 → (𝑦 ∈ 𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V))
14	11, 13	sylan9r 508	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓‘𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V))
15	14	ralrimiv 3124	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
16	10, 15	sylan2 593	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  ∪ ciun 4955  suc csuc 6334  ‘cfv 6511  ωcom 7842   ∧ w-bnj17 34676   predc-bnj14 34678   FrSe w-bnj15 34682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fv 6519  df-om 7843  df-bnj17 34677  df-bnj14 34679  df-bnj13 34681  df-bnj15 34683

This theorem is referenced by:  bnj535  34880  bnj546  34886