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Theorem fin23lem36 9114
Description: Lemma for fin23 9155. 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 6148 . . . . . . 7 (𝑎 = 𝐵 → (𝑌𝑎) = (𝑌𝐵))
21rneqd 5313 . . . . . 6 (𝑎 = 𝐵 → ran (𝑌𝑎) = ran (𝑌𝐵))
32unieqd 4412 . . . . 5 (𝑎 = 𝐵 ran (𝑌𝑎) = ran (𝑌𝐵))
43sseq1d 3611 . . . 4 (𝑎 = 𝐵 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝐵) ⊆ ran (𝑌𝐵)))
54imbi2d 330 . . 3 (𝑎 = 𝐵 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝐵) ⊆ ran (𝑌𝐵))))
6 fveq2 6148 . . . . . . 7 (𝑎 = 𝑏 → (𝑌𝑎) = (𝑌𝑏))
76rneqd 5313 . . . . . 6 (𝑎 = 𝑏 → ran (𝑌𝑎) = ran (𝑌𝑏))
87unieqd 4412 . . . . 5 (𝑎 = 𝑏 ran (𝑌𝑎) = ran (𝑌𝑏))
98sseq1d 3611 . . . 4 (𝑎 = 𝑏 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝑏) ⊆ ran (𝑌𝐵)))
109imbi2d 330 . . 3 (𝑎 = 𝑏 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝑏) ⊆ ran (𝑌𝐵))))
11 fveq2 6148 . . . . . . 7 (𝑎 = suc 𝑏 → (𝑌𝑎) = (𝑌‘suc 𝑏))
1211rneqd 5313 . . . . . 6 (𝑎 = suc 𝑏 → ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1312unieqd 4412 . . . . 5 (𝑎 = suc 𝑏 ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1413sseq1d 3611 . . . 4 (𝑎 = suc 𝑏 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵)))
1514imbi2d 330 . . 3 (𝑎 = suc 𝑏 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵))))
16 fveq2 6148 . . . . . . 7 (𝑎 = 𝐴 → (𝑌𝑎) = (𝑌𝐴))
1716rneqd 5313 . . . . . 6 (𝑎 = 𝐴 → ran (𝑌𝑎) = ran (𝑌𝐴))
1817unieqd 4412 . . . . 5 (𝑎 = 𝐴 ran (𝑌𝑎) = ran (𝑌𝐴))
1918sseq1d 3611 . . . 4 (𝑎 = 𝐴 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝐴) ⊆ ran (𝑌𝐵)))
2019imbi2d 330 . . 3 (𝑎 = 𝐴 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝐴) ⊆ ran (𝑌𝐵))))
21 ssid 3603 . . . 4 ran (𝑌𝐵) ⊆ ran (𝑌𝐵)
22212a1i 12 . . 3 (𝐵 ∈ ω → (𝜑 ran (𝑌𝐵) ⊆ ran (𝑌𝐵)))
23 simprr 795 . . . . . . . 8 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → 𝜑)
24 simpll 789 . . . . . . . 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 9113 . . . . . . . 8 ((𝜑𝑏 ∈ ω) → ran (𝑌‘suc 𝑏) ⊊ ran (𝑌𝑏))
3123, 24, 30syl2anc 692 . . . . . . 7 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊊ ran (𝑌𝑏))
3231pssssd 3682 . . . . . 6 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝑏))
33 sstr2 3590 . . . . . 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 642 . . . 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 7040 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝜑 ran (𝑌𝐴) ⊆ ran (𝑌𝐵)))
3837impr 648 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → ran (𝑌𝐴) ⊆ ran (𝑌𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3186  wss 3555  wpss 3556  𝒫 cpw 4130   cuni 4402   cint 4440  ran crn 5075  cres 5076  suc csuc 5684  1-1wf1 5844  cfv 5847  (class class class)co 6604  ωcom 7012  reccrdg 7450  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451
This theorem is referenced by: (None)
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