Theorem bnj150 30200

Description: Technical lemma for bnj151 30201. 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
bnj150.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj150.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj150.3	⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj150.4	⊢ (𝜑′ ↔ [1𝑜 / 𝑛]𝜑)
bnj150.5	⊢ (𝜓′ ↔ [1𝑜 / 𝑛]𝜓)
bnj150.6	⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)))
bnj150.7	⊢ (𝜁′ ↔ [1𝑜 / 𝑛]𝜁)
bnj150.8	⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
bnj150.9	⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
bnj150.10	⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
bnj150.11	⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)

Assertion

Ref	Expression
bnj150	⊢ 𝜃0

Distinct variable groups:   𝐴,𝑓,𝑛,𝑥   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑛,𝑥   𝑓,𝜁″   𝑖,𝑛,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁(𝑥,𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝐹(𝑥,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜑″(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓″(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜁″(𝑥,𝑦,𝑖,𝑛)   𝜃₀(𝑥,𝑦,𝑓,𝑖,𝑛)

Proof of Theorem bnj150

Step	Hyp	Ref	Expression
1		0ex 4718	. . . . . . . . . 10 ⊢ ∅ ∈ V
2		bnj93 30187	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
3		funsng 5851	. . . . . . . . . 10 ⊢ ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
4	1, 2, 3	sylancr 694	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
5		bnj150.8	. . . . . . . . . 10 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
6	5	funeqi 5824	. . . . . . . . 9 ⊢ (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
7	4, 6	sylibr 223	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹)
8	5	bnj96 30189	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1𝑜)
9	7, 8	bnj1422 30162	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 1𝑜)
10	5	bnj97 30190	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
11		bnj150.1	. . . . . . . . 9 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
12		bnj150.4	. . . . . . . . 9 ⊢ (𝜑′ ↔ [1𝑜 / 𝑛]𝜑)
13		bnj150.9	. . . . . . . . 9 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
14	11, 12, 13, 5	bnj125 30196	. . . . . . . 8 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
15	10, 14	sylibr 223	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑″)
16	9, 15	jca 553	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1𝑜 ∧ 𝜑″))
17		bnj98 30191	. . . . . . 7 ⊢ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))
18		bnj150.2	. . . . . . . 8 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
19		bnj150.5	. . . . . . . 8 ⊢ (𝜓′ ↔ [1𝑜 / 𝑛]𝜓)
20		bnj150.10	. . . . . . . 8 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
21	18, 19, 20, 5	bnj126 30197	. . . . . . 7 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1𝑜 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
22	17, 21	mpbir 220	. . . . . 6 ⊢ 𝜓″
23	16, 22	jctir 559	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 Fn 1𝑜 ∧ 𝜑″) ∧ 𝜓″))
24		df-3an 1033	. . . . 5 ⊢ ((𝐹 Fn 1𝑜 ∧ 𝜑″ ∧ 𝜓″) ↔ ((𝐹 Fn 1𝑜 ∧ 𝜑″) ∧ 𝜓″))
25	23, 24	sylibr 223	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1𝑜 ∧ 𝜑″ ∧ 𝜓″))
26		bnj150.11	. . . . 5 ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
27		bnj150.3	. . . . . 6 ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
28		bnj150.7	. . . . . 6 ⊢ (𝜁′ ↔ [1𝑜 / 𝑛]𝜁)
29	27, 28, 12, 19	bnj121 30194	. . . . 5 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)))
30	5, 13, 20, 26, 29	bnj124 30195	. . . 4 ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1𝑜 ∧ 𝜑″ ∧ 𝜓″)))
31	25, 30	mpbir 220	. . 3 ⊢ 𝜁″
32	5	bnj95 30188	. . . 4 ⊢ 𝐹 ∈ V
33		sbceq1a 3413	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝜁′ ↔ [𝐹 / 𝑓]𝜁′))
34	33, 26	syl6bbr 277	. . . 4 ⊢ (𝑓 = 𝐹 → (𝜁′ ↔ 𝜁″))
35	32, 34	spcev 3273	. . 3 ⊢ (𝜁″ → ∃𝑓𝜁′)
36	31, 35	ax-mp 5	. 2 ⊢ ∃𝑓𝜁′
37		bnj150.6	. . . 4 ⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)))
38		19.37v 1897	. . . 4 ⊢ (∃𝑓((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)))
39	37, 38	bitr4i 266	. . 3 ⊢ (𝜃0 ↔ ∃𝑓((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)))
40	39, 29	bnj133 30047	. 2 ⊢ (𝜃0 ↔ ∃𝑓𝜁′)
41	36, 40	mpbir 220	1 ⊢ 𝜃0

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  ωcom 6957  1_𝑜c1o 7440   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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-1o 7447  df-bnj13 30010  df-bnj15 30012

This theorem is referenced by:  bnj151  30201