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Theorem fin23lem36 10338
Description: Lemma for fin23 10379. 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 6887 . . . . . . 7 (𝑎 = 𝐵 → (𝑌𝑎) = (𝑌𝐵))
21rneqd 5934 . . . . . 6 (𝑎 = 𝐵 → ran (𝑌𝑎) = ran (𝑌𝐵))
32unieqd 4920 . . . . 5 (𝑎 = 𝐵 ran (𝑌𝑎) = ran (𝑌𝐵))
43sseq1d 4011 . . . 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 6887 . . . . . . 7 (𝑎 = 𝑏 → (𝑌𝑎) = (𝑌𝑏))
76rneqd 5934 . . . . . 6 (𝑎 = 𝑏 → ran (𝑌𝑎) = ran (𝑌𝑏))
87unieqd 4920 . . . . 5 (𝑎 = 𝑏 ran (𝑌𝑎) = ran (𝑌𝑏))
98sseq1d 4011 . . . 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 6887 . . . . . . 7 (𝑎 = suc 𝑏 → (𝑌𝑎) = (𝑌‘suc 𝑏))
1211rneqd 5934 . . . . . 6 (𝑎 = suc 𝑏 → ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1312unieqd 4920 . . . . 5 (𝑎 = suc 𝑏 ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1413sseq1d 4011 . . . 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 6887 . . . . . . 7 (𝑎 = 𝐴 → (𝑌𝑎) = (𝑌𝐴))
1716rneqd 5934 . . . . . 6 (𝑎 = 𝐴 → ran (𝑌𝑎) = ran (𝑌𝐴))
1817unieqd 4920 . . . . 5 (𝑎 = 𝐴 ran (𝑌𝑎) = ran (𝑌𝐴))
1918sseq1d 4011 . . . 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 4002 . . . 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 10337 . . . . . . . 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 4095 . . . . . 6 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝑏))
33 sstr2 3987 . . . . . 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 7884 . 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 3946  wpss 3947  𝒫 cpw 4600   cuni 4906   cint 4948  ran crn 5675  cres 5676  suc csuc 6362  1-1wf1 6536  cfv 6539  (class class class)co 7403  ωcom 7849  reccrdg 8403  m cmap 8815
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 5297  ax-nul 5304  ax-pr 5425  ax-un 7719
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 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7406  df-om 7850  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404
This theorem is referenced by: (None)
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