Theorem bnj535 35407

Description: Technical lemma for bnj852 35438. 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
bnj535.1	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj535.2	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj535.3	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj535.4	⊢ (𝜏 ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))

Assertion

Ref	Expression
bnj535	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)

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)   𝐴(𝑥, 𝑓, 𝑚, 𝑛)   𝑅(𝑥, 𝑓, 𝑚, 𝑛)   𝐺(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜑′(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝜓′(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)

Proof of Theorem bnj535

Step	Hyp	Ref	Expression
1		bnj422 35233	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏))
2		bnj251 35220	. . 3 ⊢ ((𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏))))
3	1, 2	bitri 278	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏))))
4		fvex 6895	. . . . . . . . 9 ⊢ (𝑓‘𝑝) ∈ V
5		bnj535.1	. . . . . . . . . 10 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
6		bnj535.2	. . . . . . . . . 10 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
7		bnj535.4	. . . . . . . . . 10 ⊢ (𝜏 ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
8	5, 6, 7	bnj518 35403	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
9		iunexg 7964	. . . . . . . . 9 ⊢ (((𝑓‘𝑝) ∈ V ∧ ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
10	4, 8, 9	sylancr 599	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
11		vex 3457	. . . . . . . . 9 ⊢ 𝑚 ∈ V
12	11	bnj519 35254	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → Fun {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
13	10, 12	syl 18	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → Fun {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
14		dmsnopg 6213	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → dom {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {𝑚})
15	10, 14	syl 18	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → dom {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {𝑚})
16	13, 15	bnj1422 35354	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚})
17		disjcsn 9586	. . . . . . 7 ⊢ (𝑚 ∩ {𝑚}) = ∅
18		fnun 6650	. . . . . . 7 ⊢ (((𝑓 Fn 𝑚 ∧ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚}) ∧ (𝑚 ∩ {𝑚}) = ∅) → (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
19	17, 18	mpan2 704	. . . . . 6 ⊢ ((𝑓 Fn 𝑚 ∧ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚}) → (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
20	16, 19	sylan2 605	. . . . 5 ⊢ ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏)) → (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
21		bnj535.3	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
22	21	fneq1i 6633	. . . . 5 ⊢ (𝐺 Fn (𝑚 ∪ {𝑚}) ↔ (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
23	20, 22	sylibr 237	. . . 4 ⊢ ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏)) → 𝐺 Fn (𝑚 ∪ {𝑚}))
24		fneq2 6628	. . . 4 ⊢ (𝑛 = (𝑚 ∪ {𝑚}) → (𝐺 Fn 𝑛 ↔ 𝐺 Fn (𝑚 ∪ {𝑚})))
25	23, 24	imbitrrid 249	. . 3 ⊢ (𝑛 = (𝑚 ∪ {𝑚}) → ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏)) → 𝐺 Fn 𝑛))
26	25	imp 412	. 2 ⊢ ((𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏))) → 𝐺 Fn 𝑛)
27	3, 26	sylbi 220	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ∪ cun 3900   ∩ cin 3901  ∅c0 4282  {csn 4587  ⟨cop 4593  ∪ ciun 4954  dom cdm 5659  suc csuc 6363  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537  ωcom 7866   ∧ w-bnj17 35204   predc-bnj14 35206   FrSe w-bnj15 35210

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740  ax-reg 9568

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-om 7867  df-bnj17 35205  df-bnj14 35207  df-bnj13 35209  df-bnj15 35211

This theorem is used by:  bnj543  35410