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Theorem bnj548 31093
Description: Technical lemma for bnj852 31117. 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)
21bnj930 30966 . . . . . 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 1096 . . . . . . 7 (𝜏𝑓 Fn 𝑚)
6 fndm 6028 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
7 eleq2 2719 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
87biimpar 501 . . . . . . . 8 ((dom 𝑓 = 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
96, 8sylan 487 . . . . . . 7 ((𝑓 Fn 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
105, 9sylan 487 . . . . . 6 ((𝜏𝑖𝑚) → 𝑖 ∈ dom 𝑓)
11103ad2antl2 1244 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝑖 ∈ dom 𝑓)
123, 11jca 553 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑖 ∈ dom 𝑓))
13 bnj548.4 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
1413bnj931 30967 . . . 4 𝑓𝐺
1512, 14jctil 559 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
16 3anan12 1069 . . 3 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
1715, 16sylibr 224 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓))
18 funssfv 6247 . 2 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) → (𝐺𝑖) = (𝑓𝑖))
19 iuneq1 4566 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2019eqcomd 2657 . . 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 2710 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  cun 3605  wss 3607  {csn 4210  cop 4216   ciun 4552  dom cdm 5143  Fun wfun 5920   Fn wfn 5921  cfv 5926   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-rab 2950  df-v 3233  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-iun 4554  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  bnj553  31094
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