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Theorem bnj548 30667
Description: Technical lemma for bnj852 30691. 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 30540 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
32adantr 481 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → Fun 𝐺)
4 bnj548.1 . . . . . . . 8 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
54simp1bi 1074 . . . . . . 7 (𝜏𝑓 Fn 𝑚)
6 fndm 5950 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
7 eleq2 2693 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
87biimpar 502 . . . . . . . 8 ((dom 𝑓 = 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
96, 8sylan 488 . . . . . . 7 ((𝑓 Fn 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
105, 9sylan 488 . . . . . 6 ((𝜏𝑖𝑚) → 𝑖 ∈ dom 𝑓)
11103ad2antl2 1222 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝑖 ∈ dom 𝑓)
123, 11jca 554 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑖 ∈ dom 𝑓))
13 bnj548.4 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
1413bnj931 30541 . . . 4 𝑓𝐺
1512, 14jctil 559 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
16 3anan12 1049 . . 3 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
1715, 16sylibr 224 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓))
18 funssfv 6167 . 2 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) → (𝐺𝑖) = (𝑓𝑖))
19 iuneq1 4505 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2019eqcomd 2632 . . 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 2685 . 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 384  w3a 1036   = wceq 1480  wcel 1992  cun 3558  wss 3560  {csn 4153  cop 4159   ciun 4490  dom cdm 5079  Fun wfun 5844   Fn wfn 5845  cfv 5850   predc-bnj14 30453   FrSe w-bnj15 30457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858
This theorem is referenced by:  bnj553  30668
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