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Theorem fin23lem35 9372
Description: Lemma for fin23 9414. 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 9371 . . . 4 ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
7 fvex 6343 . . . . . . 7 (𝑌𝐴) ∈ V
8 f1eq1 6237 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → (𝑗:ω–1-1→V ↔ (𝑌𝐴):ω–1-1→V))
9 rneq 5490 . . . . . . . . . . 11 (𝑗 = (𝑌𝐴) → ran 𝑗 = ran (𝑌𝐴))
109unieqd 4585 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → ran 𝑗 = ran (𝑌𝐴))
1110sseq1d 3782 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ( ran 𝑗𝐺 ran (𝑌𝐴) ⊆ 𝐺))
128, 11anbi12d 610 . . . . . . . 8 (𝑗 = (𝑌𝐴) → ((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) ↔ ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
13 fveq2 6333 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → (𝑖𝑗) = (𝑖‘(𝑌𝐴)))
14 f1eq1 6237 . . . . . . . . . 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 5492 . . . . . . . . . . 11 (𝑗 = (𝑌𝐴) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝐴)))
1716unieqd 4585 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝐴)))
1817, 10psseq12d 3852 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ( ran (𝑖𝑗) ⊊ ran 𝑗 ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
1915, 18anbi12d 610 . . . . . . . 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 333 . . . . . . 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 3451 . . . . . 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 466 . . . 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 479 . 2 ((𝜑𝐴 ∈ ω) → ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴))
26 frsuc 7686 . . . . . . 7 (𝐴 ∈ ω → ((rec(𝑖, ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴)))
2726adantl 467 . . . . . 6 ((𝜑𝐴 ∈ ω) → ((rec(𝑖, ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴)))
285fveq1i 6334 . . . . . 6 (𝑌‘suc 𝐴) = ((rec(𝑖, ) ↾ ω)‘suc 𝐴)
295fveq1i 6334 . . . . . . 7 (𝑌𝐴) = ((rec(𝑖, ) ↾ ω)‘𝐴)
3029fveq2i 6336 . . . . . 6 (𝑖‘(𝑌𝐴)) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴))
3127, 28, 303eqtr4g 2830 . . . . 5 ((𝜑𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌𝐴)))
3231rneqd 5492 . . . 4 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌𝐴)))
3332unieqd 4585 . . 3 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌𝐴)))
3433psseq1d 3850 . 2 ((𝜑𝐴 ∈ ω) → ( ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴) ↔ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
3525, 34mpbird 247 1 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wcel 2145  {cab 2757  wral 3061  Vcvv 3351  wss 3724  wpss 3725  𝒫 cpw 4298   cuni 4575   cint 4612  ran crn 5251  cres 5252  suc csuc 5869  1-1wf1 6029  cfv 6032  (class class class)co 6794  ωcom 7213  reccrdg 7659  𝑚 cmap 8010
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-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  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-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-om 7214  df-wrecs 7560  df-recs 7622  df-rdg 7660
This theorem is referenced by:  fin23lem36  9373  fin23lem38  9374  fin23lem39  9375
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