Theorem bnj149 32257

Description: Technical lemma for bnj151 32259. 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
bnj149.1	⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
bnj149.2	⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
bnj149.3	⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)
bnj149.4	⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
bnj149.5	⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
bnj149.6	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj149	⊢ 𝜃1

Distinct variable groups:   𝐴,𝑓,𝑔,𝑥   𝑅,𝑓,𝑔,𝑥   𝑓,𝜁₁   𝑔,𝜁₀

Allowed substitution hints:   𝜑′(𝑥,𝑓,𝑔)   𝜓′(𝑥,𝑓,𝑔)   𝜁₀(𝑥,𝑓)   𝜑₁(𝑥,𝑓,𝑔)   𝜓₁(𝑥,𝑓,𝑔)   𝜃₁(𝑥,𝑓,𝑔)   𝜁₁(𝑥,𝑔)

Proof of Theorem bnj149

Step	Hyp	Ref	Expression
1		simpr1 1191	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn 1o)
2		df1o2 8099	. . . . . . . . 9 ⊢ 1o = {∅}
3	2	fneq2i 6421	. . . . . . . 8 ⊢ (𝑓 Fn 1o ↔ 𝑓 Fn {∅})
4	1, 3	sylib 221	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn {∅})
5		simpr2 1192	. . . . . . . . . 10 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝜑′)
6		bnj149.6	. . . . . . . . . 10 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
7	5, 6	sylib 221	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8		fvex 6658	. . . . . . . . . 10 ⊢ (𝑓‘∅) ∈ V
9	8	elsn 4540	. . . . . . . . 9 ⊢ ((𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
10	7, 9	sylibr 237	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
11		0ex 5175	. . . . . . . . 9 ⊢ ∅ ∈ V
12		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (𝑓‘𝑔) = (𝑓‘∅))
13	12	eleq1d 2874	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ((𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)}))
14	11, 13	ralsn 4579	. . . . . . . 8 ⊢ (∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
15	10, 14	sylibr 237	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → ∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)})
16		ffnfv 6859	. . . . . . 7 ⊢ (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓 Fn {∅} ∧ ∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)}))
17	4, 15, 16	sylanbrc 586	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)})
18		bnj93 32245	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
19	18	adantr 484	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → pred(𝑥, 𝐴, 𝑅) ∈ V)
20		fsng 6876	. . . . . . 7 ⊢ ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
21	11, 19, 20	sylancr 590	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
22	17, 21	mpbid 235	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
23	22	ex 416	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
24	23	alrimiv 1928	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∀𝑓((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
25		mo2icl 3653	. . 3 ⊢ (∀𝑓((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
26	24, 25	syl 17	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
27		bnj149.1	. 2 ⊢ (𝜃1 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))
28	26, 27	mpbir 234	1 ⊢ 𝜃1

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∃*wmo 2596  ∀wral 3106  Vcvv 3441  [wsbc 3720  ∅c0 4243  {csn 4525  ⟨cop 4531   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  1_oc1o 8078   predc-bnj14 32068   FrSe w-bnj15 32072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-reu 3113  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1o 8085  df-bnj13 32071  df-bnj15 32073

This theorem is referenced by:  bnj151  32259