Theorem bnj580 34389

Description: Technical lemma for bnj579 34390. 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 1144	. . . . . 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 34384	. . . . . . 7 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
7	6	simp1bi 1144	. . . . . 6 ⊢ (𝜒′ → 𝑔 Fn 𝑛)
8	2, 7	bnj240 34175	. . . . 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 34354	. . . . . . . . . . . . 13 ⊢ (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
13		vex 3477	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
14	10, 4, 13	bnj540 34368	. . . . . . . . . . . . 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 34387	. . . . . . . . . . . . 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 34388	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝜏) → 𝜃)
19	18	ex 412	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑛 → (𝜏 → 𝜃))
20	19	rgen 3062	. . . . . . . . . 10 ⊢ ∀𝑗 ∈ 𝑛 (𝜏 → 𝜃)
21		vex 3477	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
22	21, 17	bnj110 34334	. . . . . . . . . 10 ⊢ (( E Fr 𝑛 ∧ ∀𝑗 ∈ 𝑛 (𝜏 → 𝜃)) → ∀𝑗 ∈ 𝑛 𝜃)
23	20, 22	mpan2 688	. . . . . . . . 9 ⊢ ( E Fr 𝑛 → ∀𝑗 ∈ 𝑛 𝜃)
24	15	ralbii 3092	. . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝑛 𝜃 ↔ ∀𝑗 ∈ 𝑛 ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
25	23, 24	sylib 217	. . . . . . . 8 ⊢ ( E Fr 𝑛 → ∀𝑗 ∈ 𝑛 ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
26	25	r19.21be 3248	. . . . . . 7 ⊢ ∀𝑗 ∈ 𝑛 ( E Fr 𝑛 → ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
27	11	bnj923 34244	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
28		nnord 7867	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → Ord 𝑛)
29		ordfr 6379	. . . . . . . . . . . . 13 ⊢ (Ord 𝑛 → E Fr 𝑛)
30	27, 28, 29	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
31	30	3ad2ant1 1132	. . . . . . . . . . 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 3092	. . . . . . 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 230	. . . . . 6 ⊢ ∀𝑗 ∈ 𝑛 ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗))
38		r19.21v 3178	. . . . . 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 229	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → ∀𝑗 ∈ 𝑛 (𝑓‘𝑗) = (𝑔‘𝑗))
40		eqfnfv 7032	. . . . . 6 ⊢ ((𝑓 Fn 𝑛 ∧ 𝑔 Fn 𝑛) → (𝑓 = 𝑔 ↔ ∀𝑗 ∈ 𝑛 (𝑓‘𝑗) = (𝑔‘𝑗)))
41	40	biimprd 247	. . . . 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 1121	. . 3 ⊢ (𝑛 ∈ 𝐷 → ((𝜒 ∧ 𝜒′) → 𝑓 = 𝑔))
44	43	alrimivv 1930	. 2 ⊢ (𝑛 ∈ 𝐷 → ∀𝑓∀𝑔((𝜒 ∧ 𝜒′) → 𝑓 = 𝑔))
45		sbsbc 3781	. . . . . 6 ⊢ ([𝑔 / 𝑓]𝜒 ↔ [𝑔 / 𝑓]𝜒)
46	45	anbi2i 622	. . . . 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 1821	. . 3 ⊢ (∀𝑓∀𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔) ↔ ∀𝑓∀𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
49		nfv 1916	. . . 4 ⊢ Ⅎ𝑔𝜒
50	49	mo3 2557	. . 3 ⊢ (∃*𝑓𝜒 ↔ ∀𝑓∀𝑔((𝜒 ∧ [𝑔 / 𝑓]𝜒) → 𝑓 = 𝑔))
51	5	anbi2i 622	. . . . 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 1821	. . 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 233	1 ⊢ (𝑛 ∈ 𝐷 → ∃*𝑓𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  [wsb 2066   ∈ wcel 2105  ∃*wmo 2531  ∀wral 3060  [wsbc 3777   ∖ cdif 3945  ∅c0 4322  {csn 4628  ∪ ciun 4997   class class class wbr 5148   E cep 5579   Fr wfr 5628  Ord word 6363  suc csuc 6366   Fn wfn 6538  ‘cfv 6543  ωcom 7859   predc-bnj14 34164

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-om 7860  df-bnj17 34163

This theorem is referenced by:  bnj579  34390