Theorem bnj546 35229

Description: Technical lemma for bnj852 35254. 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
bnj546.1	⊢ 𝐷 = (ω ∖ {∅})
bnj546.2	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj546.3	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj546.4	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj546.5	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj546	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′

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

Proof of Theorem bnj546

Step	Hyp	Ref	Expression
1		bnj546.2	. . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
2		3simpc 1166	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) → (𝜑′ ∧ 𝜓′))
3	1, 2	sylbi 220	. . . . . 6 ⊢ (𝜏 → (𝜑′ ∧ 𝜓′))
4		bnj546.3	. . . . . . 7 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
5		bnj546.1	. . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅})
6	5	bnj923 35102	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝐷 → 𝑚 ∈ ω)
7	6	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → 𝑚 ∈ ω)
8		simp3 1154	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → 𝑝 ∈ 𝑚)
9	7, 8	jca 520	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
10	4, 9	sylbi 220	. . . . . 6 ⊢ (𝜎 → (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
11	3, 10	anim12i 624	. . . . 5 ⊢ ((𝜏 ∧ 𝜎) → ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
12		bnj256 35040	. . . . 5 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
13	11, 12	sylibr 237	. . . 4 ⊢ ((𝜏 ∧ 𝜎) → (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
14	13	anim2i 628	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) → (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
15	14	3impb 1130	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
16		bnj546.4	. . 3 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
17		bnj546.5	. . 3 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
18		biid 264	. . 3 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
19	16, 17, 18	bnj518 35219	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
20		fvex 6895	. . 3 ⊢ (𝑓‘𝑝) ∈ V
21		iunexg 7960	. . 3 ⊢ (((𝑓‘𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
22	20, 21	mpan 702	. 2 ⊢ (∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
23	15, 19, 22	3syl 19	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3461   ∖ cdif 3908  ∅c0 4292  {csn 4592  ∪ ciun 4958  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7862   ∧ w-bnj17 35020   predc-bnj14 35022   FrSe w-bnj15 35026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fv 6545  df-om 7863  df-bnj17 35021  df-bnj14 35023  df-bnj13 35025  df-bnj15 35027

This theorem is referenced by:  bnj938  35270