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Theorem fin23lem35 10345
Description: Lemma for fin23 10387. 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 10344 . . . 4 ((𝜑𝐴 ∈ ω) → ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺))
7 fvex 6905 . . . . . . 7 (𝑌𝐴) ∈ V
8 f1eq1 6783 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → (𝑗:ω–1-1→V ↔ (𝑌𝐴):ω–1-1→V))
9 rneq 5936 . . . . . . . . . . 11 (𝑗 = (𝑌𝐴) → ran 𝑗 = ran (𝑌𝐴))
109unieqd 4923 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → ran 𝑗 = ran (𝑌𝐴))
1110sseq1d 4014 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ( ran 𝑗𝐺 ran (𝑌𝐴) ⊆ 𝐺))
128, 11anbi12d 630 . . . . . . . 8 (𝑗 = (𝑌𝐴) → ((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) ↔ ((𝑌𝐴):ω–1-1→V ∧ ran (𝑌𝐴) ⊆ 𝐺)))
13 fveq2 6892 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → (𝑖𝑗) = (𝑖‘(𝑌𝐴)))
14 f1eq1 6783 . . . . . . . . . 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 5938 . . . . . . . . . . 11 (𝑗 = (𝑌𝐴) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝐴)))
1716unieqd 4923 . . . . . . . . . 10 (𝑗 = (𝑌𝐴) → ran (𝑖𝑗) = ran (𝑖‘(𝑌𝐴)))
1817, 10psseq12d 4095 . . . . . . . . 9 (𝑗 = (𝑌𝐴) → ( ran (𝑖𝑗) ⊊ ran 𝑗 ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
1915, 18anbi12d 630 . . . . . . . 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 343 . . . . . . 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 3596 . . . . . 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 8440 . . . . . . 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 6893 . . . . . 6 (𝑌‘suc 𝐴) = ((rec(𝑖, ) ↾ ω)‘suc 𝐴)
295fveq1i 6893 . . . . . . 7 (𝑌𝐴) = ((rec(𝑖, ) ↾ ω)‘𝐴)
3029fveq2i 6895 . . . . . 6 (𝑖‘(𝑌𝐴)) = (𝑖‘((rec(𝑖, ) ↾ ω)‘𝐴))
3127, 28, 303eqtr4g 2796 . . . . 5 ((𝜑𝐴 ∈ ω) → (𝑌‘suc 𝐴) = (𝑖‘(𝑌𝐴)))
3231rneqd 5938 . . . 4 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌𝐴)))
3332unieqd 4923 . . 3 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) = ran (𝑖‘(𝑌𝐴)))
3433psseq1d 4093 . 2 ((𝜑𝐴 ∈ ω) → ( ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴) ↔ ran (𝑖‘(𝑌𝐴)) ⊊ ran (𝑌𝐴)))
3525, 34mpbird 256 1 ((𝜑𝐴 ∈ ω) → ran (𝑌‘suc 𝐴) ⊊ ran (𝑌𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1538   = wceq 1540  wcel 2105  {cab 2708  wral 3060  Vcvv 3473  wss 3949  wpss 3950  𝒫 cpw 4603   cuni 4909   cint 4951  ran crn 5678  cres 5679  suc csuc 6367  1-1wf1 6541  cfv 6544  (class class class)co 7412  ωcom 7858  reccrdg 8412  m cmap 8823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413
This theorem is referenced by:  fin23lem36  10346  fin23lem38  10347  fin23lem39  10348
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