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Theorem bnj548 32277
Description: Technical lemma for bnj852 32301. 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 32149 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
32adantr 484 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → Fun 𝐺)
4 bnj548.1 . . . . . . . 8 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
54simp1bi 1142 . . . . . . 7 (𝜏𝑓 Fn 𝑚)
6 fndm 6429 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
7 eleq2 2881 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
87biimpar 481 . . . . . . . 8 ((dom 𝑓 = 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
96, 8sylan 583 . . . . . . 7 ((𝑓 Fn 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
105, 9sylan 583 . . . . . 6 ((𝜏𝑖𝑚) → 𝑖 ∈ dom 𝑓)
11103ad2antl2 1183 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝑖 ∈ dom 𝑓)
123, 11jca 515 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑖 ∈ dom 𝑓))
13 bnj548.4 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
1413bnj931 32150 . . . 4 𝑓𝐺
1512, 14jctil 523 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
16 3anan12 1093 . . 3 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
1715, 16sylibr 237 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓))
18 funssfv 6670 . 2 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) → (𝐺𝑖) = (𝑓𝑖))
19 iuneq1 4900 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2019eqcomd 2807 . . 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 2861 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  cun 3882  wss 3884  {csn 4528  cop 4534   ciun 4884  dom cdm 5523  Fun wfun 6322   Fn wfn 6323  cfv 6328   predc-bnj14 32066   FrSe w-bnj15 32070
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  bnj553  32278
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