Theorem bnj150 33887

Description: Technical lemma for bnj151 33888. 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		bnj150.8	. . . 4 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
2	1	bnj95 33875	. . 3 ⊢ 𝐹 ∈ V
3		sbceq1a 3789	. . . 4 ⊢ (𝑓 = 𝐹 → (𝜁′ ↔ [𝐹 / 𝑓]𝜁′))
4		bnj150.11	. . . 4 ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
5	3, 4	bitr4di 289	. . 3 ⊢ (𝑓 = 𝐹 → (𝜁′ ↔ 𝜁″))
6		0ex 5308	. . . . . . . . 9 ⊢ ∅ ∈ V
7		bnj93 33874	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
8		funsng 6600	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
9	6, 7, 8	sylancr 588	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
10	1	funeqi 6570	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
11	9, 10	sylibr 233	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹)
12	1	bnj96 33876	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o)
13	11, 12	bnj1422 33848	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 1o)
14	1	bnj97 33877	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
15		bnj150.1	. . . . . . . 8 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
16		bnj150.4	. . . . . . . 8 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
17		bnj150.9	. . . . . . . 8 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
18	15, 16, 17, 1	bnj125 33883	. . . . . . 7 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
19	14, 18	sylibr 233	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑″)
20	13, 19	jca 513	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″))
21		bnj98 33878	. . . . . 6 ⊢ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))
22		bnj150.2	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
23		bnj150.5	. . . . . . 7 ⊢ (𝜓′ ↔ [1o / 𝑛]𝜓)
24		bnj150.10	. . . . . . 7 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′)
25	22, 23, 24, 1	bnj126 33884	. . . . . 6 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
26	21, 25	mpbir 230	. . . . 5 ⊢ 𝜓″
27		df-3an 1090	. . . . 5 ⊢ ((𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″) ↔ ((𝐹 Fn 1o ∧ 𝜑″) ∧ 𝜓″))
28	20, 26, 27	sylanblrc 591	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″))
29		bnj150.3	. . . . . 6 ⊢ (𝜁 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
30		bnj150.7	. . . . . 6 ⊢ (𝜁′ ↔ [1o / 𝑛]𝜁)
31	29, 30, 16, 23	bnj121 33881	. . . . 5 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
32	1, 17, 24, 4, 31	bnj124 33882	. . . 4 ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)))
33	28, 32	mpbir 230	. . 3 ⊢ 𝜁″
34	2, 5, 33	ceqsexv2d 3529	. 2 ⊢ ∃𝑓𝜁′
35		bnj150.6	. . . 4 ⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
36		19.37v 1996	. . . 4 ⊢ (∃𝑓((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
37	35, 36	bitr4i 278	. . 3 ⊢ (𝜃0 ↔ ∃𝑓((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
38	37, 31	bnj133 33738	. 2 ⊢ (𝜃0 ↔ ∃𝑓𝜁′)
39	34, 38	mpbir 230	1 ⊢ 𝜃0

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  [wsbc 3778  ∅c0 4323  {csn 4629  ⟨cop 4635  ∪ ciun 4998  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544  ωcom 7855  1_oc1o 8459   predc-bnj14 33699   FrSe w-bnj15 33703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-1o 8466  df-bnj13 33702  df-bnj15 33704

This theorem is referenced by:  bnj151  33888