Theorem bnj149 34867

Description: Technical lemma for bnj151 34869. 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 1193	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) → 𝑓 Fn 1o)
2		df1o2 8511	. . . . . . . . 9 ⊢ 1o = {∅}
3	2	fneq2i 6666	. . . . . . . 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 1194	. . . . . . . . . 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 6919	. . . . . . . . . 10 ⊢ (𝑓‘∅) ∈ V
9	8	elsn 4645	. . . . . . . . 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 5312	. . . . . . . . 9 ⊢ ∅ ∈ V
12		fveq2 6906	. . . . . . . . . 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 4685	. . . . . . . 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 7138	. . . . . . 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 34855	. . . . . . . 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 7156	. . . . . . 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 1924	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∀𝑓((𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
25		mo2icl 3722	. . 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 1534   = wceq 1536   ∈ wcel 2105  ∃*wmo 2535  ∀wral 3058  Vcvv 3477  [wsbc 3790  ∅c0 4338  {csn 4630  ⟨cop 4636   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  1_oc1o 8497   predc-bnj14 34680   FrSe w-bnj15 34684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-1o 8504  df-bnj13 34683  df-bnj15 34685

This theorem is referenced by:  bnj151  34869