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Theorem bnj149 32232
 Description: Technical lemma for bnj151 32234. 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 1191 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → 𝑓 Fn 1o)
2 df1o2 8114 . . . . . . . . 9 1o = {∅}
32fneq2i 6441 . . . . . . . 8 (𝑓 Fn 1o𝑓 Fn {∅})
41, 3sylib 221 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → 𝑓 Fn {∅})
5 simpr2 1192 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → 𝜑′)
6 bnj149.6 . . . . . . . . . 10 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
75, 6sylib 221 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8 fvex 6676 . . . . . . . . . 10 (𝑓‘∅) ∈ V
98elsn 4565 . . . . . . . . 9 ((𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
107, 9sylibr 237 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
11 0ex 5198 . . . . . . . . 9 ∅ ∈ V
12 fveq2 6663 . . . . . . . . . 10 (𝑔 = ∅ → (𝑓𝑔) = (𝑓‘∅))
1312eleq1d 2900 . . . . . . . . 9 (𝑔 = ∅ → ((𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)}))
1411, 13ralsn 4604 . . . . . . . 8 (∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
1510, 14sylibr 237 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → ∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)})
16 ffnfv 6875 . . . . . . 7 (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓 Fn {∅} ∧ ∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)}))
174, 15, 16sylanbrc 586 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → 𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)})
18 bnj93 32220 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
1918adantr 484 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → pred(𝑥, 𝐴, 𝑅) ∈ V)
20 fsng 6892 . . . . . . 7 ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2111, 19, 20sylancr 590 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2217, 21mpbid 235 . . . . 5 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1o𝜑′𝜓′)) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
2322ex 416 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴) → ((𝑓 Fn 1o𝜑′𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2423alrimiv 1929 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → ∀𝑓((𝑓 Fn 1o𝜑′𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
25 mo2icl 3691 . . 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 234 1 𝜃1
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2115  ∃*wmo 2622  ∀wral 3133  Vcvv 3480  [wsbc 3758  ∅c0 4276  {csn 4550  ⟨cop 4556   Fn wfn 6340  ⟶wf 6341  ‘cfv 6345  1oc1o 8093   predc-bnj14 32043   FrSe w-bnj15 32047 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-reu 3140  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-1o 8100  df-bnj13 32046  df-bnj15 32048 This theorem is referenced by:  bnj151  32234
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