Theorem bnj543 35190

Description: Technical lemma for bnj852 35218. 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 35005	. . . . . . 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 35007	. . . . . . 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 277	. . . . . 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 35002	. . . . . 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 35004	. . . . . 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 303	. . . . 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 35004	. . . . . 6 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
8	7	3anbi1i 1171	. . . . 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 34996	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚))
11	9, 10	bitri 277	. . . . . 6 ⊢ (𝜏 ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚))
12		bnj543.5	. . . . . . 7 ⊢ (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
13		3anan32 1109	. . . . . . 7 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
14	12, 13	bitri 277	. . . . . 6 ⊢ (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
15	11, 14	anbi12i 637	. . . . 5 ⊢ ((𝜏 ∧ 𝜎) ↔ (((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚)))
16	6, 8, 15	3bitr4ri 306	. . . 4 ⊢ ((𝜏 ∧ 𝜎) ↔ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
17	16	anbi2i 632	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚)))
18		3anass 1107	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)))
19		bnj252 35001	. . 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 305	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
21		df-suc 6354	. . . . . . 7 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
22	21	eqeq2i 2777	. . . . . 6 ⊢ (𝑛 = suc 𝑚 ↔ 𝑛 = (𝑚 ∪ {𝑚}))
23	22	3anbi2i 1172	. . . . 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 632	. . . 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 35001	. . . 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 305	. . 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 263	. . . 4 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
31	27, 28, 29, 30	bnj535 35187	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
32	26, 31	sylbi 219	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
33	20, 32	sylbi 219	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ∪ cun 3904  ∅c0 4287  {csn 4584  ⟨cop 4590  ∪ ciun 4951  suc csuc 6350   Fn wfn 6518  ‘cfv 6523  ωcom 7848   ∧ w-bnj17 34984   predc-bnj14 34986   FrSe w-bnj15 34990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720  ax-reg 9542

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531  df-om 7849  df-bnj17 34985  df-bnj14 34987  df-bnj13 34989  df-bnj15 34991

This theorem is referenced by:  bnj544  35191