Theorem bnj535 33932

Description: Technical lemma for bnj852 33963. 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 33757	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏))
2		bnj251 33744	. . 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 275	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏))))
4		fvex 6905	. . . . . . . . 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 33928	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∀𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
9		iunexg 7950	. . . . . . . . 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 588	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V)
11		vex 3479	. . . . . . . . 9 ⊢ 𝑚 ∈ V
12	11	bnj519 33778	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ∈ V → Fun {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
13	10, 12	syl 17	. . . . . . 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 17	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → dom {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} = {𝑚})
16	13, 15	bnj1422 33879	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏) → {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩} Fn {𝑚})
17		disjcsn 9599	. . . . . . 7 ⊢ (𝑚 ∩ {𝑚}) = ∅
18		fnun 6664	. . . . . . 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 690	. . . . . 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 594	. . . . 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 6647	. . . . 5 ⊢ (𝐺 Fn (𝑚 ∪ {𝑚}) ↔ (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩}) Fn (𝑚 ∪ {𝑚}))
23	20, 22	sylibr 233	. . . 4 ⊢ ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏)) → 𝐺 Fn (𝑚 ∪ {𝑚}))
24		fneq2 6642	. . . 4 ⊢ (𝑛 = (𝑚 ∪ {𝑚}) → (𝐺 Fn 𝑛 ↔ 𝐺 Fn (𝑚 ∪ {𝑚})))
25	23, 24	imbitrrid 245	. . 3 ⊢ (𝑛 = (𝑚 ∪ {𝑚}) → ((𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏)) → 𝐺 Fn 𝑛))
26	25	imp 408	. 2 ⊢ ((𝑛 = (𝑚 ∪ {𝑚}) ∧ (𝑓 Fn 𝑚 ∧ (𝑅 FrSe 𝐴 ∧ 𝜏))) → 𝐺 Fn 𝑛)
27	3, 26	sylbi 216	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∪ cun 3947   ∩ cin 3948  ∅c0 4323  {csn 4629  ⟨cop 4635  ∪ ciun 4998  dom cdm 5677  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544  ωcom 7855   ∧ w-bnj17 33728   predc-bnj14 33730   FrSe w-bnj15 33734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-reg 9587

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-om 7856  df-bnj17 33729  df-bnj14 33731  df-bnj13 33733  df-bnj15 33735

This theorem is referenced by:  bnj543  33935