Theorem bnj543 34886

Description: Technical lemma for bnj852 34914. 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 34700	. . . . . . 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 34702	. . . . . . 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 275	. . . . . 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 34697	. . . . . 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 34699	. . . . . 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 301	. . . . 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 34699	. . . . . 6 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
8	7	3anbi1i 1156	. . . . 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 34691	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚))
11	9, 10	bitri 275	. . . . . 6 ⊢ (𝜏 ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚))
12		bnj543.5	. . . . . . 7 ⊢ (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
13		3anan32 1096	. . . . . . 7 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
14	12, 13	bitri 275	. . . . . 6 ⊢ (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
15	11, 14	anbi12i 628	. . . . 5 ⊢ ((𝜏 ∧ 𝜎) ↔ (((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚)))
16	6, 8, 15	3bitr4ri 304	. . . 4 ⊢ ((𝜏 ∧ 𝜎) ↔ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
17	16	anbi2i 623	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚)))
18		3anass 1094	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)))
19		bnj252 34696	. . 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 303	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
21		df-suc 6392	. . . . . . 7 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
22	21	eqeq2i 2748	. . . . . 6 ⊢ (𝑛 = suc 𝑚 ↔ 𝑛 = (𝑚 ∪ {𝑚}))
23	22	3anbi2i 1157	. . . . 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 623	. . . 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 34696	. . . 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 303	. . 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 261	. . . 4 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
31	27, 28, 29, 30	bnj535 34883	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
32	26, 31	sylbi 217	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
33	20, 32	sylbi 217	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∪ cun 3961  ∅c0 4339  {csn 4631  ⟨cop 4637  ∪ ciun 4996  suc csuc 6388   Fn wfn 6558  ‘cfv 6563  ωcom 7887   ∧ w-bnj17 34679   predc-bnj14 34681   FrSe w-bnj15 34685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-reg 9630

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-om 7888  df-bnj17 34680  df-bnj14 34682  df-bnj13 34684  df-bnj15 34686

This theorem is referenced by:  bnj544  34887