Theorem bnj149 34872

Description: Technical lemma for bnj151 34874. 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 1195	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn 1o)
2		df1o2 8444	. . . . . . . . 9 ⊢ 1o = {∅}
3	2	fneq2i 6619	. . . . . . . 8 ⊢ (𝑓 Fn 1o ↔ 𝑓 Fn {∅})
4	1, 3	sylib 218	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn {∅})
5		simpr2 1196	. . . . . . . . . 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 218	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8		fvex 6874	. . . . . . . . . 10 ⊢ (𝑓‘∅) ∈ V
9	8	elsn 4607	. . . . . . . . 9 ⊢ ((𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
10	7, 9	sylibr 234	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
11		0ex 5265	. . . . . . . . 9 ⊢ ∅ ∈ V
12		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (𝑓‘𝑔) = (𝑓‘∅))
13	12	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ((𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)}))
14	11, 13	ralsn 4648	. . . . . . . 8 ⊢ (∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
15	10, 14	sylibr 234	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → ∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)})
16		ffnfv 7094	. . . . . . 7 ⊢ (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓 Fn {∅} ∧ ∀𝑔 ∈ {∅} (𝑓‘𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)}))
17	4, 15, 16	sylanbrc 583	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)})
18		bnj93 34860	. . . . . . . 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 7112	. . . . . . 7 ⊢ ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
21	11, 19, 20	sylancr 587	. . . . . 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 232	. . . . 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 1927	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∀𝑓((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
25		mo2icl 3688	. . 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 231	1 ⊢ 𝜃1

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2532  ∀wral 3045  Vcvv 3450  [wsbc 3756  ∅c0 4299  {csn 4592  ⟨cop 4598   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  1_oc1o 8430   predc-bnj14 34685   FrSe w-bnj15 34689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-1o 8437  df-bnj13 34688  df-bnj15 34690

This theorem is referenced by:  bnj151  34874