Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem36 Structured version   Visualization version   GIF version

Theorem fin23lem36 9821
 Description: Lemma for fin23 9862. 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 6663 . . . . . . 7 (𝑎 = 𝐵 → (𝑌𝑎) = (𝑌𝐵))
21rneqd 5784 . . . . . 6 (𝑎 = 𝐵 → ran (𝑌𝑎) = ran (𝑌𝐵))
32unieqd 4815 . . . . 5 (𝑎 = 𝐵 ran (𝑌𝑎) = ran (𝑌𝐵))
43sseq1d 3925 . . . 4 (𝑎 = 𝐵 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝐵) ⊆ ran (𝑌𝐵)))
54imbi2d 344 . . 3 (𝑎 = 𝐵 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝐵) ⊆ ran (𝑌𝐵))))
6 fveq2 6663 . . . . . . 7 (𝑎 = 𝑏 → (𝑌𝑎) = (𝑌𝑏))
76rneqd 5784 . . . . . 6 (𝑎 = 𝑏 → ran (𝑌𝑎) = ran (𝑌𝑏))
87unieqd 4815 . . . . 5 (𝑎 = 𝑏 ran (𝑌𝑎) = ran (𝑌𝑏))
98sseq1d 3925 . . . 4 (𝑎 = 𝑏 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝑏) ⊆ ran (𝑌𝐵)))
109imbi2d 344 . . 3 (𝑎 = 𝑏 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝑏) ⊆ ran (𝑌𝐵))))
11 fveq2 6663 . . . . . . 7 (𝑎 = suc 𝑏 → (𝑌𝑎) = (𝑌‘suc 𝑏))
1211rneqd 5784 . . . . . 6 (𝑎 = suc 𝑏 → ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1312unieqd 4815 . . . . 5 (𝑎 = suc 𝑏 ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1413sseq1d 3925 . . . 4 (𝑎 = suc 𝑏 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵)))
1514imbi2d 344 . . 3 (𝑎 = suc 𝑏 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵))))
16 fveq2 6663 . . . . . . 7 (𝑎 = 𝐴 → (𝑌𝑎) = (𝑌𝐴))
1716rneqd 5784 . . . . . 6 (𝑎 = 𝐴 → ran (𝑌𝑎) = ran (𝑌𝐴))
1817unieqd 4815 . . . . 5 (𝑎 = 𝐴 ran (𝑌𝑎) = ran (𝑌𝐴))
1918sseq1d 3925 . . . 4 (𝑎 = 𝐴 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝐴) ⊆ ran (𝑌𝐵)))
2019imbi2d 344 . . 3 (𝑎 = 𝐴 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝐴) ⊆ ran (𝑌𝐵))))
21 ssid 3916 . . . 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 9820 . . . . . . . 8 ((𝜑𝑏 ∈ ω) → ran (𝑌‘suc 𝑏) ⊊ ran (𝑌𝑏))
3123, 24, 30syl2anc 587 . . . . . . 7 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊊ ran (𝑌𝑏))
3231pssssd 4005 . . . . . 6 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝑏))
33 sstr2 3901 . . . . . 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 460 . . . 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 7615 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝜑 ran (𝑌𝐴) ⊆ ran (𝑌𝐵)))
3837impr 458 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → ran (𝑌𝐴) ⊆ ran (𝑌𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  Vcvv 3409   ⊆ wss 3860   ⊊ wpss 3861  𝒫 cpw 4497  ∪ cuni 4801  ∩ cint 4841  ran crn 5529   ↾ cres 5530  suc csuc 6176  –1-1→wf1 6337  ‘cfv 6340  (class class class)co 7156  ωcom 7585  reccrdg 8061   ↑m cmap 8422 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-om 7586  df-wrecs 7963  df-recs 8024  df-rdg 8062 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator