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Theorem fin23lem35 10387
Description: Lemma for fin23 10429. 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 10386 . . . 4 ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
7 fvex 6919 . . . . . . 7 (𝑌𝐴) ∈ V
8 f1eq1 6799 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → (𝑗:ω–1-1→V ↔ (𝑌𝐴):ω–1-1→V))
9 rneq 5947 . . . . . . . . . . 11 (𝑗 = (𝑌𝐴) → ran 𝑗 = ran (𝑌𝐴))
109unieqd 4920 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → ran 𝑗 = ran (𝑌𝐴))
1110sseq1d 4015 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ( ran 𝑗𝐺 ran (𝑌𝐴) ⊆ 𝐺))
128, 11anbi12d 632 . . . . . . . 8 (𝑗 = (𝑌𝐴) → ((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) ↔ ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
13 fveq2 6906 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → (𝑖𝑗) = (𝑖‘(𝑌𝐴)))
14 f1eq1 6799 . . . . . . . . . 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 5949 . . . . . . . . . . 11 (𝑗 = (𝑌𝐴) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝐴)))
1716unieqd 4920 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝐴)))
1817, 10psseq12d 4097 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ( ran (𝑖𝑗) ⊊ ran 𝑗 ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
1915, 18anbi12d 632 . . . . . . . 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 344 . . . . . . 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 3605 . . . . . 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 480 . . . 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 495 . 2 ((𝜑𝐴 ∈ ω) → ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴))
26 frsuc 8477 . . . . . . 7 (𝐴 ∈ ω → ((rec(𝑖, ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴)))
2726adantl 481 . . . . . 6 ((𝜑𝐴 ∈ ω) → ((rec(𝑖, ) ↾ ω)‘suc 𝐴) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴)))
285fveq1i 6907 . . . . . 6 (𝑌‘suc 𝐴) = ((rec(𝑖, ) ↾ ω)‘suc 𝐴)
295fveq1i 6907 . . . . . . 7 (𝑌𝐴) = ((rec(𝑖, ) ↾ ω)‘𝐴)
3029fveq2i 6909 . . . . . 6 (𝑖‘(𝑌𝐴)) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴))
3127, 28, 303eqtr4g 2802 . . . . 5 ((𝜑𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌𝐴)))
3231rneqd 5949 . . . 4 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌𝐴)))
3332unieqd 4920 . . 3 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌𝐴)))
3433psseq1d 4095 . 2 ((𝜑𝐴 ∈ ω) → ( ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴) ↔ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
3525, 34mpbird 257 1 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  wss 3951  wpss 3952  𝒫 cpw 4600   cuni 4907   cint 4946  ran crn 5686  cres 5687  suc csuc 6386  1-1wf1 6558  cfv 6561  (class class class)co 7431  ωcom 7887  reccrdg 8449  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  fin23lem36  10388  fin23lem38  10389  fin23lem39  10390
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