Theorem bnj543 35290

Description: Technical lemma for bnj852 35318. 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
bnj543.1	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj543.2	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj543.3	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj543.4	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj543.5	⊢ (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))

Assertion

Ref	Expression
bnj543	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 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)   𝜎(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 bnj543

Step	Hyp	Ref	Expression
1		bnj257 35105	. . . . . . 7 ⊢ (((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑓 Fn 𝑚 ∧ 𝑛 = suc 𝑚))
2		bnj268 35107	. . . . . . 7 ⊢ (((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑓 Fn 𝑚 ∧ 𝑛 = suc 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
3	1, 2	bitri 278	. . . . . 6 ⊢ (((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
4		bnj253 35102	. . . . . 6 ⊢ (((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) ↔ (((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
5		bnj256 35104	. . . . . 6 ⊢ (((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚 ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚) ↔ (((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚)))
6	3, 4, 5	3bitr3i 304	. . . . 5 ⊢ ((((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) ↔ (((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚)))
7		bnj256 35104	. . . . . 6 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
8	7	3anbi1i 1174	. . . . 5 ⊢ (((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) ↔ (((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
9		bnj543.4	. . . . . . 7 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
10		bnj170 35096	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚))
11	9, 10	bitri 278	. . . . . 6 ⊢ (𝜏 ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚))
12		bnj543.5	. . . . . . 7 ⊢ (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
13		3anan32 1112	. . . . . . 7 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
14	12, 13	bitri 278	. . . . . 6 ⊢ (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
15	11, 14	anbi12i 639	. . . . 5 ⊢ ((𝜏 ∧ 𝜎) ↔ (((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚)))
16	6, 8, 15	3bitr4ri 307	. . . 4 ⊢ ((𝜏 ∧ 𝜎) ↔ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
17	16	anbi2i 634	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚)))
18		3anass 1110	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)))
19		bnj252 35101	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚)))
20	17, 18, 19	3bitr4i 306	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
21		df-suc 6366	. . . . . . 7 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
22	21	eqeq2i 2775	. . . . . 6 ⊢ (𝑛 = suc 𝑚 ↔ 𝑛 = (𝑚 ∪ {𝑚}))
23	22	3anbi2i 1175	. . . . 5 ⊢ (((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) ↔ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
24	23	anbi2i 634	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚)))
25		bnj252 35101	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚)))
26	24, 19, 25	3bitr4i 306	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚))
27		bnj543.1	. . . 4 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
28		bnj543.2	. . . 4 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
29		bnj543.3	. . . 4 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
30		biid 264	. . . 4 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
31	27, 28, 29, 30	bnj535 35287	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
32	26, 31	sylbi 220	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
33	20, 32	sylbi 220	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078   ∪ cun 3902  ∅c0 4285  {csn 4588  ⟨cop 4594  ∪ ciun 4955  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7860   ∧ w-bnj17 35084   predc-bnj14 35086   FrSe w-bnj15 35090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-om 7861  df-bnj17 35085  df-bnj14 35087  df-bnj13 35089  df-bnj15 35091

This theorem is used by:  bnj544  35291