Theorem bnj580 34944

Description: Technical lemma for bnj579 34945. 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
bnj580.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj580.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj580.3	⊢ (𝜒 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj580.4	⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑)
bnj580.5	⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓)
bnj580.6	⊢ (𝜒′ ↔ [𝑔 / 𝑓]𝜒)
bnj580.7	⊢ 𝐷 = (ω ∖ {∅})
bnj580.8	⊢ (𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
bnj580.9	⊢ (𝜏 ↔ ∀𝑘 ∈ 𝑛 (𝑘 E 𝑗 → [𝑘 / 𝑗]𝜃))

Assertion

Ref	Expression
bnj580	⊢ (𝑛 ∈ 𝐷 → ∃*𝑓𝜒)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑘   𝐷,𝑓,𝑔,𝑗,𝑘   𝑅,𝑓,𝑖,𝑘   𝜒,𝑔,𝑗,𝑘   𝑗,𝜒′,𝑘   𝑓,𝑛   𝑔,𝑖,𝑛,𝑘   𝑥,𝑓   𝑦,𝑓,𝑔,𝑖,𝑘   𝑗,𝑛   𝜃,𝑘

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜒(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜃(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝜏(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝐴(𝑥,𝑦,𝑔,𝑗,𝑛)   𝐷(𝑥,𝑦,𝑖,𝑛)   𝑅(𝑥,𝑦,𝑔,𝑗,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜒′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑛)

Proof of Theorem bnj580

Step	Hyp	Ref	Expression
1		bnj580.3	. . . . . . 7 ⊢ (𝜒 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2	1	simp1bi 1145	. . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛)
3		bnj580.4	. . . . . . . 8 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑)
4		bnj580.5	. . . . . . . 8 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓)
5		bnj580.6	. . . . . . . 8 ⊢ (𝜒′ ↔ [𝑔 / 𝑓]𝜒)
6	1, 3, 4, 5	bnj581 34939	. . . . . . 7 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
7	6	simp1bi 1145	. . . . . 6 ⊢ (𝜒′ → 𝑔 Fn 𝑛)
8	2, 7	bnj240 34730	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓 Fn 𝑛 ∧ 𝑔 Fn 𝑛))
9		bnj580.1	. . . . . . . . . . . . 13 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
10		bnj580.2	. . . . . . . . . . . . 13 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
11		bnj580.7	. . . . . . . . . . . . 13 ⊢ 𝐷 = (ω ∖ {∅})
12	3, 9	bnj154 34909	. . . . . . . . . . . . 13 ⊢ (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
13		vex 3463	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
14	10, 4, 13	bnj540 34923	. . . . . . . . . . . . 13 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑔‘suc 𝑖) = ∪ 𝑦 ∈ (𝑔‘𝑖) pred(𝑦, 𝐴, 𝑅)))
15		bnj580.8	. . . . . . . . . . . . 13 ⊢ (𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
16	15	bnj591 34942	. . . . . . . . . . . . 13 ⊢ ([𝑘 / 𝑗]𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))
17		bnj580.9	. . . . . . . . . . . . 13 ⊢ (𝜏 ↔ ∀𝑘 ∈ 𝑛 (𝑘 E 𝑗 → [𝑘 / 𝑗]𝜃))
18	9, 10, 1, 11, 12, 14, 6, 15, 16, 17	bnj594 34943	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝜏) → 𝜃)
19	18	ex 412	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑛 → (𝜏 → 𝜃))
20	19	rgen 3053	. . . . . . . . . 10 ⊢ ∀𝑗 ∈ 𝑛 (𝜏 → 𝜃)
21		vex 3463	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
22	21, 17	bnj110 34889	. . . . . . . . . 10 ⊢ (( E Fr 𝑛 ∧ ∀𝑗 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑗 ∈ 𝑛 𝜃)
23	20, 22	mpan2 691	. . . . . . . . 9 ⊢ ( E Fr 𝑛 → ∀𝑗 ∈ 𝑛 𝜃)
24	15	ralbii 3082	. . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝑛 𝜃 ↔ ∀𝑗 ∈ 𝑛 ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
25	23, 24	sylib 218	. . . . . . . 8 ⊢ ( E Fr 𝑛 → ∀𝑗 ∈ 𝑛 ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
26	25	r19.21be 3235	. . . . . . 7 ⊢ ∀𝑗 ∈ 𝑛 ( E Fr 𝑛 → ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
27	11	bnj923 34799	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
28		nnord 7869	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → Ord 𝑛)
29		ordfr 6367	. . . . . . . . . . . . 13 ⊢ (Ord 𝑛 → E Fr 𝑛)
30	27, 28, 29	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
31	30	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → E Fr 𝑛)
32	31	pm4.71ri 560	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) ↔ ( E Fr 𝑛 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′)))
33	32	imbi1i 349	. . . . . . . . 9 ⊢ (((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ (( E Fr 𝑛 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′)) → (𝑓‘𝑗) = (𝑔‘𝑗)))
34		impexp 450	. . . . . . . . 9 ⊢ ((( E Fr 𝑛 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′)) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ( E Fr 𝑛 → ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗))))
35	33, 34	bitri 275	. . . . . . . 8 ⊢ (((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ( E Fr 𝑛 → ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗))))
36	35	ralbii 3082	. . . . . . 7 ⊢ (∀𝑗 ∈ 𝑛 ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ∀𝑗 ∈ 𝑛 ( E Fr 𝑛 → ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗))))
37	26, 36	mpbir 231	. . . . . 6 ⊢ ∀𝑗 ∈ 𝑛 ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗))
38		r19.21v 3165	. . . . . 6 ⊢ (∀𝑗 ∈ 𝑛 ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → ∀𝑗 ∈ 𝑛 (𝑓‘𝑗) = (𝑔‘𝑗)))
39	37, 38	mpbi 230	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → ∀𝑗 ∈ 𝑛 (𝑓‘𝑗) = (𝑔‘𝑗))
40		eqfnfv 7021	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑔 Fn 𝑛) → (𝑓 = 𝑔 ↔ ∀𝑗 ∈ 𝑛 (𝑓‘𝑗) = (𝑔‘𝑗)))
41	40	biimprd 248	. . . . 5 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑔 Fn 𝑛) → (∀𝑗 ∈ 𝑛 (𝑓‘𝑗) = (𝑔‘𝑗) → 𝑓 = 𝑔))
42	8, 39, 41	sylc 65	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → 𝑓 = 𝑔)
43	42	3expib 1122	. . 3 ⊢ (𝑛 ∈ 𝐷 → ((𝜒 ∧ 𝜒′) → 𝑓 = 𝑔))
44	43	alrimivv 1928	. 2 ⊢ (𝑛 ∈ 𝐷 → ∀𝑓∀𝑔((𝜒 ∧ 𝜒′) → 𝑓 = 𝑔))
45		sbsbc 3769	. . . . . 6 ⊢ ([𝑔 / 𝑓]𝜒 ↔ [𝑔 / 𝑓]𝜒)
46	45	anbi2i 623	. . . . 5 ⊢ ((𝜒 ∧ [𝑔 / 𝑓]𝜒) ↔ (𝜒 ∧ [𝑔 / 𝑓]𝜒))
47	46	imbi1i 349	. . . 4 ⊢ (((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
48	47	2albii 1820	. . 3 ⊢ (∀𝑓∀𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ∀𝑓∀𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
49		nfv 1914	. . . 4 ⊢ Ⅎ𝑔𝜒
50	49	mo3 2563	. . 3 ⊢ (∃*𝑓𝜒 ↔ ∀𝑓∀𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
51	5	anbi2i 623	. . . . 5 ⊢ ((𝜒 ∧ 𝜒′) ↔ (𝜒 ∧ [𝑔 / 𝑓]𝜒))
52	51	imbi1i 349	. . . 4 ⊢ (((𝜒 ∧ 𝜒′) → 𝑓 = 𝑔) ↔ ((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
53	52	2albii 1820	. . 3 ⊢ (∀𝑓∀𝑔((𝜒 ∧ 𝜒′) → 𝑓 = 𝑔) ↔ ∀𝑓∀𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
54	48, 50, 53	3bitr4i 303	. 2 ⊢ (∃*𝑓𝜒 ↔ ∀𝑓∀𝑔((𝜒 ∧ 𝜒′) → 𝑓 = 𝑔))
55	44, 54	sylibr 234	1 ⊢ (𝑛 ∈ 𝐷 → ∃*𝑓𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  [wsb 2064   ∈ wcel 2108  ∃*wmo 2537  ∀wral 3051  [wsbc 3765   ∖ cdif 3923  ∅c0 4308  {csn 4601  ∪ ciun 4967   class class class wbr 5119   E cep 5552   Fr wfr 5603  Ord word 6351  suc csuc 6354   Fn wfn 6526  ‘cfv 6531  ωcom 7861   predc-bnj14 34719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539  df-om 7862  df-bnj17 34718

This theorem is referenced by:  bnj579  34945