Theorem bnj849 30249

Description: Technical lemma for bnj69 30332. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj849.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj849.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj849.3	⊢ 𝐷 = (ω ∖ {∅})
bnj849.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj849.5	⊢ (𝜒 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
bnj849.6	⊢ (𝜃 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj849.7	⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑)
bnj849.8	⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓)
bnj849.9	⊢ (𝜃′ ↔ [𝑔 / 𝑓]𝜃)
bnj849.10	⊢ (𝜏 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))

Assertion

Ref	Expression
bnj849	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐵 ∈ V)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐵,𝑔   𝐷,𝑓,𝑔,𝑛   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑛   𝜒,𝑓,𝑔   𝜑,𝑔   𝜓,𝑔   𝜏,𝑔,𝑛   𝜃,𝑔

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

Proof of Theorem bnj849

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj849.10	. 2 ⊢ (𝜏 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
2		bnj849.1	. . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3		bnj849.2	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4		bnj849.3	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
5		bnj849.5	. . . 4 ⊢ (𝜒 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
6		bnj849.6	. . . 4 ⊢ (𝜃 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
7	2, 3, 4, 5, 6	bnj865 30247	. . 3 ⊢ ∃𝑤∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃)
8		bnj849.4	. . . . . . . 8 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9		bnj849.7	. . . . . . . 8 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑)
10		bnj849.8	. . . . . . . 8 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓)
11	8, 9, 10	bnj873 30248	. . . . . . 7 ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)}
12		df-rex 2902	. . . . . . . . 9 ⊢ (∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)))
13		19.29 1789	. . . . . . . . . . 11 ⊢ ((∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) → ∃𝑛((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ (𝑛 ∈ 𝐷 ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))))
14		an12 834	. . . . . . . . . . . . 13 ⊢ (((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ (𝑛 ∈ 𝐷 ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) ↔ (𝑛 ∈ 𝐷 ∧ ((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))))
15		df-3an 1033	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷))
16	1	anbi1i 727	. . . . . . . . . . . . . . . 16 ⊢ ((𝜏 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷))
17	15, 5, 16	3bitr4i 291	. . . . . . . . . . . . . . 15 ⊢ (𝜒 ↔ (𝜏 ∧ 𝑛 ∈ 𝐷))
18		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝜒 → 𝜒)
19		bnj849.9	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜃′ ↔ [𝑔 / 𝑓]𝜃)
20	6, 9, 10, 19	bnj581 30232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜃′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
21	19, 20	bitr3i 265	. . . . . . . . . . . . . . . . . . 19 ⊢ ([𝑔 / 𝑓]𝜃 ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
22	2, 3, 4, 5, 6	bnj864 30246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜒 → ∃!𝑓𝜃)
23		df-rex 2902	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑓 ∈ 𝑤 𝜃 ↔ ∃𝑓(𝑓 ∈ 𝑤 ∧ 𝜃))
24		exancom 1774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑓(𝑓 ∈ 𝑤 ∧ 𝜃) ↔ ∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤))
25	23, 24	sylbb 208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑓 ∈ 𝑤 𝜃 → ∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤))
26		nfeu1 2468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑓∃!𝑓𝜃
27		nfe1 2014	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑓∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤)
28	26, 27	nfan 1816	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑓(∃!𝑓𝜃 ∧ ∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤))
29		nfsbc1v 3422	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜃
30		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑓 𝑔 ∈ 𝑤
31	29, 30	nfim 1813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑓([𝑔 / 𝑓]𝜃 → 𝑔 ∈ 𝑤)
32	28, 31	nfim 1813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑓((∃!𝑓𝜃 ∧ ∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤)) → ([𝑔 / 𝑓]𝜃 → 𝑔 ∈ 𝑤))
33		sbceq1a 3413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑔 → (𝜃 ↔ [𝑔 / 𝑓]𝜃))
34		elequ1 1984	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝑤 ↔ 𝑔 ∈ 𝑤))
35	33, 34	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑔 → ((𝜃 → 𝑓 ∈ 𝑤) ↔ ([𝑔 / 𝑓]𝜃 → 𝑔 ∈ 𝑤)))
36	35	imbi2d 329	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → (((∃!𝑓𝜃 ∧ ∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤)) → (𝜃 → 𝑓 ∈ 𝑤)) ↔ ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤)) → ([𝑔 / 𝑓]𝜃 → 𝑔 ∈ 𝑤))))
37		eupick 2524	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤)) → (𝜃 → 𝑓 ∈ 𝑤))
38	32, 36, 37	chvar 2250	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∃!𝑓𝜃 ∧ ∃𝑓(𝜃 ∧ 𝑓 ∈ 𝑤)) → ([𝑔 / 𝑓]𝜃 → 𝑔 ∈ 𝑤))
39	22, 25, 38	syl2an 493	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜒 ∧ ∃𝑓 ∈ 𝑤 𝜃) → ([𝑔 / 𝑓]𝜃 → 𝑔 ∈ 𝑤))
40	21, 39	syl5bir 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜒 ∧ ∃𝑓 ∈ 𝑤 𝜃) → ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) → 𝑔 ∈ 𝑤))
41	40	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝜒 → (∃𝑓 ∈ 𝑤 𝜃 → ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) → 𝑔 ∈ 𝑤)))
42	18, 41	embantd 57	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 → ((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) → ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) → 𝑔 ∈ 𝑤)))
43	42	impd 446	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → (((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) → 𝑔 ∈ 𝑤))
44	17, 43	sylbir 224	. . . . . . . . . . . . . 14 ⊢ ((𝜏 ∧ 𝑛 ∈ 𝐷) → (((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) → 𝑔 ∈ 𝑤))
45	44	expimpd 627	. . . . . . . . . . . . 13 ⊢ (𝜏 → ((𝑛 ∈ 𝐷 ∧ ((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) → 𝑔 ∈ 𝑤))
46	14, 45	syl5bi 231	. . . . . . . . . . . 12 ⊢ (𝜏 → (((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ (𝑛 ∈ 𝐷 ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) → 𝑔 ∈ 𝑤))
47	46	exlimdv 1848	. . . . . . . . . . 11 ⊢ (𝜏 → (∃𝑛((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ (𝑛 ∈ 𝐷 ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) → 𝑔 ∈ 𝑤))
48	13, 47	syl5 33	. . . . . . . . . 10 ⊢ (𝜏 → ((∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ∧ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) → 𝑔 ∈ 𝑤))
49	48	expdimp 452	. . . . . . . . 9 ⊢ ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃)) → (∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) → 𝑔 ∈ 𝑤))
50	12, 49	syl5bi 231	. . . . . . . 8 ⊢ ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃)) → (∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) → 𝑔 ∈ 𝑤))
51	50	abssdv 3639	. . . . . . 7 ⊢ ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃)) → {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} ⊆ 𝑤)
52	11, 51	syl5eqss 3612	. . . . . 6 ⊢ ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃)) → 𝐵 ⊆ 𝑤)
53		vex 3176	. . . . . . 7 ⊢ 𝑤 ∈ V
54	53	ssex 4730	. . . . . 6 ⊢ (𝐵 ⊆ 𝑤 → 𝐵 ∈ V)
55	52, 54	syl 17	. . . . 5 ⊢ ((𝜏 ∧ ∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃)) → 𝐵 ∈ V)
56	55	ex 449	. . . 4 ⊢ (𝜏 → (∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃) → 𝐵 ∈ V))
57	56	exlimdv 1848	. . 3 ⊢ (𝜏 → (∃𝑤∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃) → 𝐵 ∈ V))
58	7, 57	mpi 20	. 2 ⊢ (𝜏 → 𝐵 ∈ V)
59	1, 58	sylbir 224	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐵 ∈ V)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173  [wsbc 3402   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ ciun 4455  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957   predc-bnj14 30007   FrSe w-bnj15 30011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012

This theorem is referenced by:  bnj893  30252