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Theorem bnj149 31324
Description: Technical lemma for bnj151 31326. 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 1248 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝑓 Fn 1𝑜)
2 df1o2 7776 . . . . . . . . 9 1𝑜 = {∅}
32fneq2i 6163 . . . . . . . 8 (𝑓 Fn 1𝑜𝑓 Fn {∅})
41, 3sylib 209 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝑓 Fn {∅})
5 simpr2 1250 . . . . . . . . . 10 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝜑′)
6 bnj149.6 . . . . . . . . . 10 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
75, 6sylib 209 . . . . . . . . 9 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8 fvex 6387 . . . . . . . . . 10 (𝑓‘∅) ∈ V
98elsn 4348 . . . . . . . . 9 ((𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
107, 9sylibr 225 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
11 0ex 4949 . . . . . . . . 9 ∅ ∈ V
12 fveq2 6374 . . . . . . . . . 10 (𝑔 = ∅ → (𝑓𝑔) = (𝑓‘∅))
1312eleq1d 2828 . . . . . . . . 9 (𝑔 = ∅ → ((𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)}))
1411, 13ralsn 4378 . . . . . . . 8 (∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓‘∅) ∈ { pred(𝑥, 𝐴, 𝑅)})
1510, 14sylibr 225 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → ∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)})
16 ffnfv 6577 . . . . . . 7 (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ (𝑓 Fn {∅} ∧ ∀𝑔 ∈ {∅} (𝑓𝑔) ∈ { pred(𝑥, 𝐴, 𝑅)}))
174, 15, 16sylanbrc 578 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)})
18 bnj93 31312 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
1918adantr 472 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → pred(𝑥, 𝐴, 𝑅) ∈ V)
20 fsng 6594 . . . . . . 7 ((∅ ∈ V ∧ pred(𝑥, 𝐴, 𝑅) ∈ V) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2111, 19, 20sylancr 581 . . . . . 6 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → (𝑓:{∅}⟶{ pred(𝑥, 𝐴, 𝑅)} ↔ 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2217, 21mpbid 223 . . . . 5 (((𝑅 FrSe 𝐴𝑥𝐴) ∧ (𝑓 Fn 1𝑜𝜑′𝜓′)) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩})
2322ex 401 . . . 4 ((𝑅 FrSe 𝐴𝑥𝐴) → ((𝑓 Fn 1𝑜𝜑′𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
2423alrimiv 2022 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → ∀𝑓((𝑓 Fn 1𝑜𝜑′𝜓′) → 𝑓 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}))
25 mo2icl 3543 . . 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 222 1 𝜃1
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  ∃*wmo 2562  wral 3054  Vcvv 3349  [wsbc 3595  c0 4078  {csn 4333  cop 4339   Fn wfn 6062  wf 6063  cfv 6067  1𝑜c1o 7756   predc-bnj14 31136   FrSe w-bnj15 31140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-1o 7763  df-bnj13 31139  df-bnj15 31141
This theorem is referenced by:  bnj151  31326
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