Theorem bnj149 32755

Description: Technical lemma for bnj151 32757. 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 1192	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn 1o)
2		df1o2 8279	. . . . . . . . 9 ⊢ 1o = {∅}
3	2	fneq2i 6515	. . . . . . . 8 ⊢ (𝑓 Fn 1o ↔ 𝑓 Fn {∅})
4	1, 3	sylib 217	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn {∅})
5		simpr2 1193	. . . . . . . . . 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 217	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8		fvex 6769	. . . . . . . . . 10 ⊢ (𝑓‘∅) ∈ V
9	8	elsn 4573	. . . . . . . . 9 ⊢ ((𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
10	7, 9	sylibr 233	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
11		0ex 5226	. . . . . . . . 9 ⊢ ∅ ∈ V
12		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (𝑓‘𝑔) = (𝑓‘∅))
13	12	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ((𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)}))
14	11, 13	ralsn 4614	. . . . . . . 8 ⊢ (∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
15	10, 14	sylibr 233	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → ∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)})
16		ffnfv 6974	. . . . . . 7 ⊢ (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓 Fn {∅} ∧ ∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)}))
17	4, 15, 16	sylanbrc 582	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)})
18		bnj93 32743	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
19	18	adantr 480	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → pred(𝑥, 𝐴, 𝑅) ∈ V)
20		fsng 6991	. . . . . . 7 ⊢ ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
21	11, 19, 20	sylancr 586	. . . . . 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 231	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
23	22	ex 412	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
24	23	alrimiv 1931	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∀𝑓((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
25		mo2icl 3644	. . 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 230	1 ⊢ 𝜃1

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  ∀wral 3063  Vcvv 3422  [wsbc 3711  ∅c0 4253  {csn 4558  ⟨cop 4564   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  1_oc1o 8260   predc-bnj14 32567   FrSe w-bnj15 32571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-1o 8267  df-bnj13 32570  df-bnj15 32572

This theorem is referenced by:  bnj151  32757