Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj545 Structured version   Visualization version   GIF version

Theorem bnj545 30708
Description: Technical lemma for bnj852 30734. 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
bnj545.1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj545.2 𝐷 = (ω ∖ {∅})
bnj545.3 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj545.4 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
bnj545.5 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj545.6 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
bnj545.7 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj545 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)

Proof of Theorem bnj545
StepHypRef Expression
1 bnj545.4 . . . . . . . . . 10 (𝜏 ↔ (𝑓 Fn 𝑚𝜑′𝜓′))
21simp1bi 1074 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
3 bnj545.5 . . . . . . . . . 10 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43simp1bi 1074 . . . . . . . . 9 (𝜎𝑚𝐷)
52, 4anim12i 589 . . . . . . . 8 ((𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
653adant1 1077 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
7 bnj545.2 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
87bnj529 30554 . . . . . . . 8 (𝑚𝐷 → ∅ ∈ 𝑚)
9 fndm 5953 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
10 eleq2 2687 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑚))
1110biimparc 504 . . . . . . . 8 ((∅ ∈ 𝑚 ∧ dom 𝑓 = 𝑚) → ∅ ∈ dom 𝑓)
128, 9, 11syl2anr 495 . . . . . . 7 ((𝑓 Fn 𝑚𝑚𝐷) → ∅ ∈ dom 𝑓)
136, 12syl 17 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → ∅ ∈ dom 𝑓)
14 bnj545.6 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
1514bnj930 30583 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
1613, 15jca 554 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜎) → (∅ ∈ dom 𝑓 ∧ Fun 𝐺))
17 bnj545.3 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
1817bnj931 30584 . . . . 5 𝑓𝐺
1916, 18jctil 559 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
20 df-3an 1038 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺))
21 3anrot 1041 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
22 ancom 466 . . . . 5 (((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2320, 21, 223bitr3i 290 . . . 4 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2419, 23sylibr 224 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
25 funssfv 6171 . . 3 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) → (𝐺‘∅) = (𝑓‘∅))
2624, 25syl 17 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝐺‘∅) = (𝑓‘∅))
271simp2bi 1075 . . 3 (𝜏𝜑′)
28273ad2ant2 1081 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑′)
29 bnj545.1 . . . 4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
30 eqtr 2640 . . . 4 (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3129, 30sylan2b 492 . . 3 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
32 bnj545.7 . . 3 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3331, 32sylibr 224 . 2 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → 𝜑″)
3426, 28, 33syl2anc 692 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cdif 3556  cun 3557  wss 3559  c0 3896  {csn 4153  cop 4159   ciun 4490  dom cdm 5079  suc csuc 5689  Fun wfun 5846   Fn wfn 5847  cfv 5852  ωcom 7019   predc-bnj14 30496   FrSe w-bnj15 30500
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-om 7020
This theorem is referenced by:  bnj600  30732  bnj908  30744
  Copyright terms: Public domain W3C validator