Theorem bnj546 32776

Description: Technical lemma for bnj852 32801. 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 1148	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) → (𝜑′ ∧ 𝜓′))
3	1, 2	sylbi 216	. . . . . 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 32648	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝐷 → 𝑚 ∈ ω)
7	6	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → 𝑚 ∈ ω)
8		simp3 1136	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → 𝑝 ∈ 𝑚)
9	7, 8	jca 511	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) → (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
10	4, 9	sylbi 216	. . . . . 6 ⊢ (𝜎 → (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
11	3, 10	anim12i 612	. . . . 5 ⊢ ((𝜏 ∧ 𝜎) → ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
12		bnj256 32585	. . . . 5 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
13	11, 12	sylibr 233	. . . 4 ⊢ ((𝜏 ∧ 𝜎) → (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
14	13	anim2i 616	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) → (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
15	14	3impb 1113	. 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 260	. . 3 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
19	16, 17, 18	bnj518 32766	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
20		fvex 6769	. . 3 ⊢ (𝑓‘𝑝) ∈ V
21		iunexg 7779	. . 3 ⊢ (((𝑓‘𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
22	20, 21	mpan 686	. 2 ⊢ (∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
23	15, 19, 22	3syl 18	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∖ cdif 3880  ∅c0 4253  {csn 4558  ∪ ciun 4921  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  ωcom 7687   ∧ w-bnj17 32565   predc-bnj14 32567   FrSe w-bnj15 32571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-bnj17 32566  df-bnj14 32568  df-bnj13 32570  df-bnj15 32572

This theorem is referenced by:  bnj938  32817