Theorem bnj518 35083

Description: Technical lemma for bnj852 35118. 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 34911	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑛 ∈ ω ∧ 𝑝 ∈ 𝑛) ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛))
3	1, 2	bitri 277	. . 3 ⊢ (𝜏 ↔ (𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛))
4		df-bnj17 34885	. . . 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 35082	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑝 ∈ 𝑛 (𝑓‘𝑝) ⊆ 𝐴)
8	7	r19.21bi 3233	. . . 4 ⊢ (((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴)
9	4, 8	sylbi 219	. . 3 ⊢ ((𝑛 ∈ ω ∧ 𝜑 ∧ 𝜓 ∧ 𝑝 ∈ 𝑛) → (𝑓‘𝑝) ⊆ 𝐴)
10	3, 9	sylbi 219	. 2 ⊢ (𝜏 → (𝑓‘𝑝) ⊆ 𝐴)
11		ssel 3911	. . . 4 ⊢ ((𝑓‘𝑝) ⊆ 𝐴 → (𝑦 ∈ (𝑓‘𝑝) → 𝑦 ∈ 𝐴))
12		bnj93 35060	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴) → pred(𝑦, 𝐴, 𝑅) ∈ V)
13	12	ex 414	. . . 4 ⊢ (𝑅 FrSe 𝐴 → (𝑦 ∈ 𝐴 → pred(𝑦, 𝐴, 𝑅) ∈ V))
14	11, 13	sylan9r 514	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → (𝑦 ∈ (𝑓‘𝑝) → pred(𝑦, 𝐴, 𝑅) ∈ V))
15	14	ralrimiv 3132	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝑓‘𝑝) ⊆ 𝐴) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
16	10, 15	sylan2 600	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ⊆ wss 3885  ∅c0 4264  ∪ ciun 4924  suc csuc 6316  ‘cfv 6489  ωcom 7810   ∧ w-bnj17 34884   predc-bnj14 34886   FrSe w-bnj15 34890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-tr 5183  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fv 6497  df-om 7811  df-bnj17 34885  df-bnj14 34887  df-bnj13 34889  df-bnj15 34891

This theorem is referenced by:  bnj535  35087  bnj546  35093