Theorem bnj543 32167

Description: Technical lemma for bnj852 32195. 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 31979	. . . . . . 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 31981	. . . . . . 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 31976	. . . . . 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 31978	. . . . . 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 31978	. . . . . 6 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
8	7	3anbi1i 1153	. . . . 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 31970	. . . . . . 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 1093	. . . . . . 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 628	. . . . 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 624	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚)))
18		3anass 1091	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)))
19		bnj252 31975	. . 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 6199	. . . . . . 7 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
22	21	eqeq2i 2836	. . . . . 6 ⊢ (𝑛 = suc 𝑚 ↔ 𝑛 = (𝑚 ∪ {𝑚}))
23	22	3anbi2i 1154	. . . . 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 624	. . . 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 31975	. . . 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 32164	. . 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 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ∪ cun 3936  ∅c0 4293  {csn 4569  ⟨cop 4575  ∪ ciun 4921  suc csuc 6195   Fn wfn 6352  ‘cfv 6357  ωcom 7582   ∧ w-bnj17 31958   predc-bnj14 31960   FrSe w-bnj15 31964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463  ax-reg 9058

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-bnj17 31959  df-bnj14 31961  df-bnj13 31963  df-bnj15 31965

This theorem is referenced by:  bnj544  32168