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Theorem bnj149 31071
Description: Technical lemma for bnj151 31073. 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 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
bnj149.2 (𝜁0 ↔ (𝑓 Fn 1𝑜𝜑′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
StepHypRef Expression
1 simpr1 1087 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝑓 Fn 1𝑜)
2 df1o2 7617 . . . . . . . . 9 1𝑜 = {∅}
32fneq2i 6024 . . . . . . . 8 (𝑓 Fn 1𝑜𝑓 Fn {∅})
41, 3sylib 208 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝑓 Fn {∅})
5 simpr2 1088 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝜑′)
6 bnj149.6 . . . . . . . . . 10 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
75, 6sylib 208 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8 fvex 6239 . . . . . . . . . 10 (𝑓‘∅) ∈ V
98elsn 4225 . . . . . . . . 9 ((𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
107, 9sylibr 224 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
11 0ex 4823 . . . . . . . . 9 ∅ ∈ V
12 fveq2 6229 . . . . . . . . . 10 (𝑔 = ∅ → (𝑓𝑔) = (𝑓‘∅))
1312eleq1d 2715 . . . . . . . . 9 (𝑔 = ∅ → ((𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)}))
1411, 13ralsn 4254 . . . . . . . 8 (∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
1510, 14sylibr 224 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → ∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)})
16 ffnfv 6428 . . . . . . 7 (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓 Fn {∅} ∧ ∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)}))
174, 15, 16sylanbrc 699 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)})
18 bnj93 31059 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
1918adantr 480 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → pred(𝑥, 𝐴, 𝑅) ∈ V)
20 fsng 6444 . . . . . . 7 ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2111, 19, 20sylancr 696 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2217, 21mpbid 222 . . . . 5 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
2322ex 449 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴) → ((𝑓 Fn 1𝑜𝜑′𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2423alrimiv 1895 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → ∀𝑓((𝑓 Fn 1𝑜𝜑′𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
25 mo2icl 3418 . . 3 (∀𝑓((𝑓 Fn 1𝑜𝜑′𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}) → ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′))
2624, 25syl 17 . 2 ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′))
27 bnj149.1 . 2 (𝜃1 ↔ ((𝑅 FrSe 𝐴𝑥𝐴) → ∃*𝑓(𝑓 Fn 1𝑜𝜑′𝜓′)))
2826, 27mpbir 221 1 𝜃1
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  ∃*wmo 2499  wral 2941  Vcvv 3231  [wsbc 3468  c0 3948  {csn 4210  cop 4216   Fn wfn 5921  wf 5922  cfv 5926  1𝑜c1o 7598   predc-bnj14 30882   FrSe w-bnj15 30886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-bnj13 30885  df-bnj15 30887
This theorem is referenced by:  bnj151  31073
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