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Theorem fin23lem36 10343
Description: Lemma for fin23 10384. Weak 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
fin23lem36 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → ran (𝑌𝐴) ⊆ ran (𝑌𝐵))
Distinct variable groups:   𝑔,𝑎,𝑖,𝑗,𝑥   𝐴,𝑎,𝑗   ,𝑎,𝐺,𝑔,𝑖,𝑗,𝑥   𝐵,𝑎   𝐹,𝑎   𝜑,𝑎,𝑗   𝑌,𝑎,𝑗
Allowed substitution hints:   𝜑(𝑥,𝑔,,𝑖)   𝐴(𝑥,𝑔,,𝑖)   𝐵(𝑥,𝑔,,𝑖,𝑗)   𝐹(𝑥,𝑔,,𝑖,𝑗)   𝑌(𝑥,𝑔,,𝑖)

Proof of Theorem fin23lem36
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . . . 7 (𝑎 = 𝐵 → (𝑌𝑎) = (𝑌𝐵))
21rneqd 5938 . . . . . 6 (𝑎 = 𝐵 → ran (𝑌𝑎) = ran (𝑌𝐵))
32unieqd 4923 . . . . 5 (𝑎 = 𝐵 ran (𝑌𝑎) = ran (𝑌𝐵))
43sseq1d 4014 . . . 4 (𝑎 = 𝐵 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝐵) ⊆ ran (𝑌𝐵)))
54imbi2d 341 . . 3 (𝑎 = 𝐵 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝐵) ⊆ ran (𝑌𝐵))))
6 fveq2 6892 . . . . . . 7 (𝑎 = 𝑏 → (𝑌𝑎) = (𝑌𝑏))
76rneqd 5938 . . . . . 6 (𝑎 = 𝑏 → ran (𝑌𝑎) = ran (𝑌𝑏))
87unieqd 4923 . . . . 5 (𝑎 = 𝑏 ran (𝑌𝑎) = ran (𝑌𝑏))
98sseq1d 4014 . . . 4 (𝑎 = 𝑏 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝑏) ⊆ ran (𝑌𝐵)))
109imbi2d 341 . . 3 (𝑎 = 𝑏 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝑏) ⊆ ran (𝑌𝐵))))
11 fveq2 6892 . . . . . . 7 (𝑎 = suc 𝑏 → (𝑌𝑎) = (𝑌‘suc 𝑏))
1211rneqd 5938 . . . . . 6 (𝑎 = suc 𝑏 → ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1312unieqd 4923 . . . . 5 (𝑎 = suc 𝑏 ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1413sseq1d 4014 . . . 4 (𝑎 = suc 𝑏 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵)))
1514imbi2d 341 . . 3 (𝑎 = suc 𝑏 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵))))
16 fveq2 6892 . . . . . . 7 (𝑎 = 𝐴 → (𝑌𝑎) = (𝑌𝐴))
1716rneqd 5938 . . . . . 6 (𝑎 = 𝐴 → ran (𝑌𝑎) = ran (𝑌𝐴))
1817unieqd 4923 . . . . 5 (𝑎 = 𝐴 ran (𝑌𝑎) = ran (𝑌𝐴))
1918sseq1d 4014 . . . 4 (𝑎 = 𝐴 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝐴) ⊆ ran (𝑌𝐵)))
2019imbi2d 341 . . 3 (𝑎 = 𝐴 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝐴) ⊆ ran (𝑌𝐵))))
21 ssid 4005 . . . 4 ran (𝑌𝐵) ⊆ ran (𝑌𝐵)
22212a1i 12 . . 3 (𝐵 ∈ ω → (𝜑 ran (𝑌𝐵) ⊆ ran (𝑌𝐵)))
23 simprr 772 . . . . . . . 8 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → 𝜑)
24 simpll 766 . . . . . . . 8 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → 𝑏 ∈ ω)
25 fin23lem33.f . . . . . . . . 9 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
26 fin23lem.f . . . . . . . . 9 (𝜑:ω–1-1→V)
27 fin23lem.g . . . . . . . . 9 (𝜑 ran 𝐺)
28 fin23lem.h . . . . . . . . 9 (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
29 fin23lem.i . . . . . . . . 9 𝑌 = (rec(𝑖, ) ↾ ω)
3025, 26, 27, 28, 29fin23lem35 10342 . . . . . . . 8 ((𝜑𝑏 ∈ ω) → ran (𝑌‘suc 𝑏) ⊊ ran (𝑌𝑏))
3123, 24, 30syl2anc 585 . . . . . . 7 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊊ ran (𝑌𝑏))
3231pssssd 4098 . . . . . 6 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝑏))
33 sstr2 3990 . . . . . 6 ( ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝑏) → ( ran (𝑌𝑏) ⊆ ran (𝑌𝐵) → ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵)))
3432, 33syl 17 . . . . 5 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ( ran (𝑌𝑏) ⊆ ran (𝑌𝐵) → ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵)))
3534expr 458 . . . 4 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → (𝜑 → ( ran (𝑌𝑏) ⊆ ran (𝑌𝐵) → ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵))))
3635a2d 29 . . 3 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝑏) → ((𝜑 ran (𝑌𝑏) ⊆ ran (𝑌𝐵)) → (𝜑 ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵))))
375, 10, 15, 20, 22, 36findsg 7890 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝜑 ran (𝑌𝐴) ⊆ ran (𝑌𝐵)))
3837impr 456 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → ran (𝑌𝐴) ⊆ ran (𝑌𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3062  Vcvv 3475  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 7409  ωcom 7855  reccrdg 8409  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410
This theorem is referenced by: (None)
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