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Theorem bnj548 34890
Description: Technical lemma for bnj852 34914. 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.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj548.1 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj548.2 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
bnj548.3 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
bnj548.4 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj548.5 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Assertion
Ref Expression
bnj548 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝐵 = 𝐾)
Distinct variable groups:   𝑦,𝐺   𝑦,𝑓   𝑦,𝑖
Allowed substitution hints:   𝜏(𝑦,𝑓,𝑖,𝑚,𝑛)   𝜎(𝑦,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑦,𝑓,𝑖,𝑚,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑚,𝑛)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛)   𝑅(𝑦,𝑓,𝑖,𝑚,𝑛)   𝐺(𝑓,𝑖,𝑚,𝑛)   𝐾(𝑦,𝑓,𝑖,𝑚,𝑛)   𝜑′(𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛)

Proof of Theorem bnj548
StepHypRef Expression
1 bnj548.5 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
21fnfund 6670 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
32adantr 480 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → Fun 𝐺)
4 bnj548.1 . . . . . . . 8 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
54simp1bi 1144 . . . . . . 7 (𝜏𝑓 Fn 𝑚)
6 fndm 6672 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
7 eleq2 2828 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
87biimpar 477 . . . . . . . 8 ((dom 𝑓 = 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
96, 8sylan 580 . . . . . . 7 ((𝑓 Fn 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
105, 9sylan 580 . . . . . 6 ((𝜏𝑖𝑚) → 𝑖 ∈ dom 𝑓)
11103ad2antl2 1185 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝑖 ∈ dom 𝑓)
123, 11jca 511 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑖 ∈ dom 𝑓))
13 bnj548.4 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
1413bnj931 34763 . . . 4 𝑓𝐺
1512, 14jctil 519 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
16 3anan12 1095 . . 3 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
1715, 16sylibr 234 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓))
18 funssfv 6928 . 2 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) → (𝐺𝑖) = (𝑓𝑖))
19 iuneq1 5013 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2019eqcomd 2741 . . 3 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅))
21 bnj548.2 . . 3 𝐵 = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
22 bnj548.3 . . 3 𝐾 = 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅)
2320, 21, 223eqtr4g 2800 . 2 ((𝐺𝑖) = (𝑓𝑖) → 𝐵 = 𝐾)
2417, 18, 233syl 18 1 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝐵 = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cun 3961  wss 3963  {csn 4631  cop 4637   ciun 4996  dom cdm 5689  Fun wfun 6557   Fn wfn 6558  cfv 6563   predc-bnj14 34681   FrSe w-bnj15 34685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  bnj553  34891
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