Theorem bnj543 31411

Description: Technical lemma for bnj852 31439. 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 31224	. . . . . . 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 31226	. . . . . . 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 266	. . . . . 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 31221	. . . . . 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 31223	. . . . . 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 292	. . . . 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 31223	. . . . . 6 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ ((𝜑′ ∧ 𝜓′) ∧ (𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚)))
8	7	3anbi1i 1196	. . . . 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 31215	. . . . . . 7 ⊢ ((𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′) ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚))
11	9, 10	bitri 266	. . . . . 6 ⊢ (𝜏 ↔ ((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚))
12		bnj543.5	. . . . . . 7 ⊢ (𝜎 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
13		3anan32 1118	. . . . . . 7 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚) ↔ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
14	12, 13	bitri 266	. . . . . 6 ⊢ (𝜎 ↔ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚))
15	11, 14	anbi12i 620	. . . . 5 ⊢ ((𝜏 ∧ 𝜎) ↔ (((𝜑′ ∧ 𝜓′) ∧ 𝑓 Fn 𝑚) ∧ ((𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚)))
16	6, 8, 15	3bitr4ri 295	. . . 4 ⊢ ((𝜏 ∧ 𝜎) ↔ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
17	16	anbi2i 616	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)) ↔ (𝑅 FrSe 𝐴 ∧ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚)))
18		3anass 1116	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ 𝜎)))
19		bnj252 31220	. . 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 294	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ↔ (𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚))
21		df-suc 5914	. . . . . . 7 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
22	21	eqeq2i 2777	. . . . . 6 ⊢ (𝑛 = suc 𝑚 ↔ 𝑛 = (𝑚 ∪ {𝑚}))
23	22	3anbi2i 1197	. . . . 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 616	. . . 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 31220	. . . 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 294	. . 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 252	. . . 4 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚))
31	27, 28, 29, 30	bnj535 31408	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = (𝑚 ∪ {𝑚}) ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
32	26, 31	sylbi 208	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ (𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚) ∧ 𝑛 = suc 𝑚 ∧ 𝑓 Fn 𝑚) → 𝐺 Fn 𝑛)
33	20, 32	sylbi 208	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  ∀wral 3055   ∪ cun 3730  ∅c0 4079  {csn 4334  ⟨cop 4340  ∪ ciun 4676  suc csuc 5910   Fn wfn 6063  ‘cfv 6068  ωcom 7263   ∧ w-bnj17 31203   predc-bnj14 31205   FrSe w-bnj15 31209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147  ax-reg 8704

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-bnj17 31204  df-bnj14 31206  df-bnj13 31208  df-bnj15 31210

This theorem is referenced by:  bnj544  31412