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Theorem fin23lem36 9053
 Description: Lemma for fin23 9094. 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 6103 . . . . . . 7 (𝑎 = 𝐵 → (𝑌𝑎) = (𝑌𝐵))
21rneqd 5274 . . . . . 6 (𝑎 = 𝐵 → ran (𝑌𝑎) = ran (𝑌𝐵))
32unieqd 4382 . . . . 5 (𝑎 = 𝐵 ran (𝑌𝑎) = ran (𝑌𝐵))
43sseq1d 3595 . . . 4 (𝑎 = 𝐵 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝐵) ⊆ ran (𝑌𝐵)))
54imbi2d 329 . . 3 (𝑎 = 𝐵 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝐵) ⊆ ran (𝑌𝐵))))
6 fveq2 6103 . . . . . . 7 (𝑎 = 𝑏 → (𝑌𝑎) = (𝑌𝑏))
76rneqd 5274 . . . . . 6 (𝑎 = 𝑏 → ran (𝑌𝑎) = ran (𝑌𝑏))
87unieqd 4382 . . . . 5 (𝑎 = 𝑏 ran (𝑌𝑎) = ran (𝑌𝑏))
98sseq1d 3595 . . . 4 (𝑎 = 𝑏 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝑏) ⊆ ran (𝑌𝐵)))
109imbi2d 329 . . 3 (𝑎 = 𝑏 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝑏) ⊆ ran (𝑌𝐵))))
11 fveq2 6103 . . . . . . 7 (𝑎 = suc 𝑏 → (𝑌𝑎) = (𝑌‘suc 𝑏))
1211rneqd 5274 . . . . . 6 (𝑎 = suc 𝑏 → ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1312unieqd 4382 . . . . 5 (𝑎 = suc 𝑏 ran (𝑌𝑎) = ran (𝑌‘suc 𝑏))
1413sseq1d 3595 . . . 4 (𝑎 = suc 𝑏 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵)))
1514imbi2d 329 . . 3 (𝑎 = suc 𝑏 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝐵))))
16 fveq2 6103 . . . . . . 7 (𝑎 = 𝐴 → (𝑌𝑎) = (𝑌𝐴))
1716rneqd 5274 . . . . . 6 (𝑎 = 𝐴 → ran (𝑌𝑎) = ran (𝑌𝐴))
1817unieqd 4382 . . . . 5 (𝑎 = 𝐴 ran (𝑌𝑎) = ran (𝑌𝐴))
1918sseq1d 3595 . . . 4 (𝑎 = 𝐴 → ( ran (𝑌𝑎) ⊆ ran (𝑌𝐵) ↔ ran (𝑌𝐴) ⊆ ran (𝑌𝐵)))
2019imbi2d 329 . . 3 (𝑎 = 𝐴 → ((𝜑 ran (𝑌𝑎) ⊆ ran (𝑌𝐵)) ↔ (𝜑 ran (𝑌𝐴) ⊆ ran (𝑌𝐵))))
21 ssid 3587 . . . 4 ran (𝑌𝐵) ⊆ ran (𝑌𝐵)
22212a1i 12 . . 3 (𝐵 ∈ ω → (𝜑 ran (𝑌𝐵) ⊆ ran (𝑌𝐵)))
23 simprr 792 . . . . . . . 8 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → 𝜑)
24 simpll 786 . . . . . . . 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 9052 . . . . . . . 8 ((𝜑𝑏 ∈ ω) → ran (𝑌‘suc 𝑏) ⊊ ran (𝑌𝑏))
3123, 24, 30syl2anc 691 . . . . . . 7 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊊ ran (𝑌𝑏))
3231pssssd 3666 . . . . . 6 (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝑏𝜑)) → ran (𝑌‘suc 𝑏) ⊆ ran (𝑌𝑏))
33 sstr2 3575 . . . . . 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 641 . . . 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 6985 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵𝐴) → (𝜑 ran (𝑌𝐴) ⊆ ran (𝑌𝐵)))
3837impr 647 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵𝐴𝜑)) → ran (𝑌𝐴) ⊆ ran (𝑌𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540   ⊊ wpss 3541  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  ran crn 5039   ↾ cres 5040  suc csuc 5642  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  ωcom 6957  reccrdg 7392   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393 This theorem is referenced by: (None)
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