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Theorem bnj545 32160
Description: Technical lemma for bnj852 32186. 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 1140 . . . . . . . . 9 (𝜏𝑓 Fn 𝑚)
3 bnj545.5 . . . . . . . . . 10 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43simp1bi 1140 . . . . . . . . 9 (𝜎𝑚𝐷)
52, 4anim12i 614 . . . . . . . 8 ((𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
653adant1 1125 . . . . . . 7 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓 Fn 𝑚𝑚𝐷))
7 bnj545.2 . . . . . . . . 9 𝐷 = (ω ∖ {∅})
87bnj529 32005 . . . . . . . 8 (𝑚𝐷 → ∅ ∈ 𝑚)
9 fndm 6448 . . . . . . . 8 (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
10 eleq2 2899 . . . . . . . . 9 (dom 𝑓 = 𝑚 → (∅ ∈ dom 𝑓 ↔ ∅ ∈ 𝑚))
1110biimparc 482 . . . . . . . 8 ((∅ ∈ 𝑚 ∧ dom 𝑓 = 𝑚) → ∅ ∈ dom 𝑓)
128, 9, 11syl2anr 598 . . . . . . 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 32034 . . . . . 6 ((𝑅 FrSe 𝐴𝜏𝜎) → Fun 𝐺)
1613, 15jca 514 . . . . 5 ((𝑅 FrSe 𝐴𝜏𝜎) → (∅ ∈ dom 𝑓 ∧ Fun 𝐺))
17 bnj545.3 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑚, 𝑦 ∈ (𝑓𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
1817bnj931 32035 . . . . 5 𝑓𝐺
1916, 18jctil 522 . . . 4 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
20 df-3an 1084 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ ((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺))
21 3anrot 1095 . . . . 5 ((∅ ∈ dom 𝑓 ∧ Fun 𝐺𝑓𝐺) ↔ (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
22 ancom 463 . . . . 5 (((∅ ∈ dom 𝑓 ∧ Fun 𝐺) ∧ 𝑓𝐺) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2320, 21, 223bitr3i 303 . . . 4 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) ↔ (𝑓𝐺 ∧ (∅ ∈ dom 𝑓 ∧ Fun 𝐺)))
2419, 23sylibr 236 . . 3 ((𝑅 FrSe 𝐴𝜏𝜎) → (Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓))
25 funssfv 6684 . . 3 ((Fun 𝐺𝑓𝐺 ∧ ∅ ∈ dom 𝑓) → (𝐺‘∅) = (𝑓‘∅))
2624, 25syl 17 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → (𝐺‘∅) = (𝑓‘∅))
271simp2bi 1141 . . 3 (𝜏𝜑′)
28273ad2ant2 1129 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑′)
29 bnj545.1 . . . 4 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
30 eqtr 2839 . . . 4 (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3129, 30sylan2b 595 . . 3 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
32 bnj545.7 . . 3 (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
3331, 32sylibr 236 . 2 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑′) → 𝜑″)
3426, 28, 33syl2anc 586 1 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  cdif 3931  cun 3932  wss 3934  c0 4289  {csn 4559  cop 4565   ciun 4910  dom cdm 5548  suc csuc 6186  Fun wfun 6342   Fn wfn 6343  cfv 6348  ωcom 7572   predc-bnj14 31951   FrSe w-bnj15 31955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-om 7573
This theorem is referenced by:  bnj600  32184  bnj908  32196
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