Theorem bnj150 35173

Description: Technical lemma for bnj151 35174. 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   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,𝑓,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 35161	. . 3 ⊢ 𝐹 ∈ V
3		sbceq1a 3757	. . . 4 ⊢ (𝑓 = 𝐹 → (𝜁′ ↔ [𝐹 / 𝑓]𝜁′))
4		bnj150.11	. . . 4 ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′)
5	3, 4	bitr4di 291	. . 3 ⊢ (𝑓 = 𝐹 → (𝜁′ ↔ 𝜁″))
6		0ex 5259	. . . . . . . . 9 ⊢ ∅ ∈ V
7		bnj93 35160	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
8		funsng 6574	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
9	6, 7, 8	sylancr 596	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
10	1	funeqi 6544	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ Fun {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
11	9, 10	sylibr 236	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹)
12	1	bnj96 35162	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → dom 𝐹 = 1o)
13	11, 12	bnj1422 35134	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝐹 Fn 1o)
14	1	bnj97 35163	. . . . . . 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 35169	. . . . . . 7 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
19	14, 18	sylibr 236	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑″)
20	13, 19	jca 519	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″))
21		bnj98 35164	. . . . . 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 35170	. . . . . 6 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
26	21, 25	mpbir 233	. . . . 5 ⊢ 𝜓″
27		df-3an 1101	. . . . 5 ⊢ ((𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″) ↔ ((𝐹 Fn 1o ∧ 𝜑″) ∧ 𝜓″))
28	20, 26, 27	sylanblrc 599	. . . 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 35167	. . . . 5 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
32	1, 17, 24, 4, 31	bnj124 35168	. . . 4 ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)))
33	28, 32	mpbir 233	. . 3 ⊢ 𝜁″
34	2, 5, 33	ceqsexv2d 3505	. 2 ⊢ ∃𝑓𝜁′
35		bnj150.6	. . . 4 ⊢ (𝜃0 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
36		19.37v 2019	. . . 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 280	. . 3 ⊢ (𝜃0 ↔ ∃𝑓((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
38	37, 31	bnj133 35025	. 2 ⊢ (𝜃0 ↔ ∃𝑓𝜁′)
39	34, 38	mpbir 233	1 ⊢ 𝜃0

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∀wral 3078  Vcvv 3456  [wsbc 3746  ∅c0 4287  {csn 4584  ⟨cop 4590  ∪ ciun 4951  suc csuc 6350  Fun wfun 6517   Fn wfn 6518  ‘cfv 6523  ωcom 7848  1_oc1o 8432   predc-bnj14 34986   FrSe w-bnj15 34990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531  df-1o 8439  df-bnj13 34989  df-bnj15 34991

This theorem is referenced by:  bnj151  35174