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Theorem fin23lem35 9765
 Description: Lemma for fin23 9807. Strict order property of 𝑌. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem33.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.f (𝜑:ω–1-1→V)
fin23lem.g (𝜑 ran 𝐺)
fin23lem.h (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
fin23lem.i 𝑌 = (rec(𝑖, ) ↾ ω)
Assertion
Ref Expression
fin23lem35 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴))
Distinct variable groups:   𝑔,𝑎,𝑖,𝑗,𝑥   𝐴,𝑎,𝑗   ,𝑎,𝐺,𝑔,𝑖,𝑗,𝑥   𝐹,𝑎   𝜑,𝑎,𝑗   𝑌,𝑎,𝑗
Allowed substitution hints:   𝜑(𝑥,𝑔,,𝑖)   𝐴(𝑥,𝑔,,𝑖)   𝐹(𝑥,𝑔,,𝑖,𝑗)   𝑌(𝑥,𝑔,,𝑖)

Proof of Theorem fin23lem35
StepHypRef Expression
1 fin23lem33.f . . . . 5 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
2 fin23lem.f . . . . 5 (𝜑:ω–1-1→V)
3 fin23lem.g . . . . 5 (𝜑 ran 𝐺)
4 fin23lem.h . . . . 5 (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
5 fin23lem.i . . . . 5 𝑌 = (rec(𝑖, ) ↾ ω)
61, 2, 3, 4, 5fin23lem34 9764 . . . 4 ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
7 fvex 6663 . . . . . . 7 (𝑌𝐴) ∈ V
8 f1eq1 6547 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → (𝑗:ω–1-1→V ↔ (𝑌𝐴):ω–1-1→V))
9 rneq 5771 . . . . . . . . . . 11 (𝑗 = (𝑌𝐴) → ran 𝑗 = ran (𝑌𝐴))
109unieqd 4815 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → ran 𝑗 = ran (𝑌𝐴))
1110sseq1d 3946 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ( ran 𝑗𝐺 ran (𝑌𝐴) ⊆ 𝐺))
128, 11anbi12d 633 . . . . . . . 8 (𝑗 = (𝑌𝐴) → ((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) ↔ ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
13 fveq2 6650 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → (𝑖𝑗) = (𝑖‘(𝑌𝐴)))
14 f1eq1 6547 . . . . . . . . . 10 ((𝑖𝑗) = (𝑖‘(𝑌𝐴)) → ((𝑖𝑗):ω–1-1→V ↔ (𝑖‘(𝑌𝐴)):ω–1-1→V))
1513, 14syl 17 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ((𝑖𝑗):ω–1-1→V ↔ (𝑖‘(𝑌𝐴)):ω–1-1→V))
1613rneqd 5773 . . . . . . . . . . 11 (𝑗 = (𝑌𝐴) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝐴)))
1716unieqd 4815 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝐴)))
1817, 10psseq12d 4022 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ( ran (𝑖𝑗) ⊊ ran 𝑗 ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
1915, 18anbi12d 633 . . . . . . . 8 (𝑗 = (𝑌𝐴) → (((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗) ↔ ((𝑖‘(𝑌𝐴)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴))))
2012, 19imbi12d 348 . . . . . . 7 (𝑗 = (𝑌𝐴) → (((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)) ↔ (((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌𝐴)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))))
217, 20spcv 3554 . . . . . 6 (∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)) → (((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌𝐴)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴))))
224, 21syl 17 . . . . 5 (𝜑 → (((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌𝐴)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴))))
2322adantr 484 . . . 4 ((𝜑𝐴 ∈ ω) → (((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺) → ((𝑖‘(𝑌𝐴)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴))))
246, 23mpd 15 . . 3 ((𝜑𝐴 ∈ ω) → ((𝑖‘(𝑌𝐴)):ω–1-1→V ∧ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
2524simprd 499 . 2 ((𝜑𝐴 ∈ ω) → ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴))
26 frsuc 8062 . . . . . . 7 (𝐴 ∈ ω → ((rec(𝑖, ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴)))
2726adantl 485 . . . . . 6 ((𝜑𝐴 ∈ ω) → ((rec(𝑖, ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴)))
285fveq1i 6651 . . . . . 6 (𝑌‘suc 𝐴) = ((rec(𝑖, ) ↾ ω)‘suc 𝐴)
295fveq1i 6651 . . . . . . 7 (𝑌𝐴) = ((rec(𝑖, ) ↾ ω)‘𝐴)
3029fveq2i 6653 . . . . . 6 (𝑖‘(𝑌𝐴)) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴))
3127, 28, 303eqtr4g 2858 . . . . 5 ((𝜑𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌𝐴)))
3231rneqd 5773 . . . 4 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌𝐴)))
3332unieqd 4815 . . 3 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌𝐴)))
3433psseq1d 4020 . 2 ((𝜑𝐴 ∈ ω) → ( ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴) ↔ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
3525, 34mpbird 260 1 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441   ⊆ wss 3881   ⊊ wpss 3882  𝒫 cpw 4497  ∪ cuni 4801  ∩ cint 4839  ran crn 5521   ↾ cres 5522  suc csuc 6164  –1-1→wf1 6324  ‘cfv 6327  (class class class)co 7140  ωcom 7567  reccrdg 8035   ↑m cmap 8396 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-om 7568  df-wrecs 7937  df-recs 7998  df-rdg 8036 This theorem is referenced by:  fin23lem36  9766  fin23lem38  9767  fin23lem39  9768
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