Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj548 Structured version   Visualization version   GIF version

Theorem bnj548 31305
Description: Technical lemma for bnj852 31329. 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 31178 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
32adantr 466 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → Fun 𝐺)
4 bnj548.1 . . . . . . . 8 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
54simp1bi 1139 . . . . . . 7 (𝜏𝑓 Fn 𝑚)
6 fndm 6130 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
7 eleq2 2839 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓𝑖𝑚))
87biimpar 463 . . . . . . . 8 ((dom 𝑓 = 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
96, 8sylan 569 . . . . . . 7 ((𝑓 Fn 𝑚𝑖𝑚) → 𝑖 ∈ dom 𝑓)
105, 9sylan 569 . . . . . 6 ((𝜏𝑖𝑚) → 𝑖 ∈ dom 𝑓)
11103ad2antl2 1201 . . . . 5 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → 𝑖 ∈ dom 𝑓)
123, 11jca 501 . . . 4 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑖 ∈ dom 𝑓))
13 bnj548.4 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
1413bnj931 31179 . . . 4 𝑓𝐺
1512, 14jctil 509 . . 3 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
16 3anan12 1081 . . 3 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (Fun 𝐺𝑖 ∈ dom 𝑓)))
1715, 16sylibr 224 . 2 (((𝑅 FrSe 𝐴𝜏𝜎) ∧ 𝑖𝑚) → (Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓))
18 funssfv 6350 . 2 ((Fun 𝐺𝑓𝐺𝑖 ∈ dom 𝑓) → (𝐺𝑖) = (𝑓𝑖))
19 iuneq1 4668 . . . 4 ((𝐺𝑖) = (𝑓𝑖) → 𝑦 ∈ (𝐺𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
2019eqcomd 2777 . . 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 2830 . 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 382  w3a 1071   = wceq 1631  wcel 2145  cun 3721  wss 3723  {csn 4316  cop 4322   ciun 4654  dom cdm 5249  Fun wfun 6025   Fn wfn 6026  cfv 6031   predc-bnj14 31094   FrSe w-bnj15 31098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  bnj553  31306
  Copyright terms: Public domain W3C validator