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 31511
Description: Technical lemma for bnj852 31537. 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 1181 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
3 bnj545.5 . . . . . . . . . 10 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43simp1bi 1181 . . . . . . . . 9 (𝜎𝑚𝐷)
52, 4anim12i 608 . . . . . . . 8 ((𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
653adant1 1166 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
7 bnj545.2 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
87bnj529 31357 . . . . . . . 8 (𝑚𝐷 → ∅ ∈ 𝑚)
9 fndm 6223 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
10 eleq2 2895 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑚))
1110biimparc 473 . . . . . . . 8 ((∅ ∈ 𝑚 ∧ dom 𝑓 = 𝑚) → ∅ ∈ dom 𝑓)
128, 9, 11syl2anr 592 . . . . . . 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 31386 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
1613, 15jca 509 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜎) → (∅ ∈ dom 𝑓 ∧ Fun 𝐺))
17 bnj545.3 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
1817bnj931 31387 . . . . 5 𝑓𝐺
1916, 18jctil 517 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
20 df-3an 1115 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺))
21 3anrot 1128 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
22 ancom 454 . . . . 5 (((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2320, 21, 223bitr3i 293 . . . 4 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2419, 23sylibr 226 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
25 funssfv 6454 . . 3 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) → (𝐺‘∅) = (𝑓‘∅))
2624, 25syl 17 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝐺‘∅) = (𝑓‘∅))
271simp2bi 1182 . . 3 (𝜏𝜑′)
28273ad2ant2 1170 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑′)
29 bnj545.1 . . . 4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
30 eqtr 2846 . . . 4 (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3129, 30sylan2b 589 . . 3 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
32 bnj545.7 . . 3 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3331, 32sylibr 226 . 2 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → 𝜑″)
3426, 28, 33syl2anc 581 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  cdif 3795  cun 3796  wss 3798  c0 4144  {csn 4397  cop 4403   ciun 4740  dom cdm 5342  suc csuc 5965  Fun wfun 6117   Fn wfn 6118  cfv 6123  ωcom 7326   predc-bnj14 31303   FrSe w-bnj15 31307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-res 5354  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-om 7327
This theorem is referenced by:  bnj600  31535  bnj908  31547
  Copyright terms: Public domain W3C validator