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Theorem wemaplem3 9571
Description: Lemma for wemapso 9574. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.)
Hypotheses
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
wemaplem2.p (𝜑𝑃 ∈ (𝐵m 𝐴))
wemaplem2.x (𝜑𝑋 ∈ (𝐵m 𝐴))
wemaplem2.q (𝜑𝑄 ∈ (𝐵m 𝐴))
wemaplem2.r (𝜑𝑅 Or 𝐴)
wemaplem2.s (𝜑𝑆 Po 𝐵)
wemaplem3.px (𝜑𝑃𝑇𝑋)
wemaplem3.xq (𝜑𝑋𝑇𝑄)
Assertion
Ref Expression
wemaplem3 (𝜑𝑃𝑇𝑄)
Distinct variable groups:   𝑥,𝐵   𝑥,𝑤,𝑦,𝑧,𝑋   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝑃,𝑥,𝑦,𝑧   𝑤,𝑄,𝑥,𝑦,𝑧   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐵(𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wemaplem3
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wemaplem3.px . . 3 (𝜑𝑃𝑇𝑋)
2 wemaplem2.p . . . 4 (𝜑𝑃 ∈ (𝐵m 𝐴))
3 wemaplem2.x . . . 4 (𝜑𝑋 ∈ (𝐵m 𝐴))
4 wemapso.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
54wemaplem1 9569 . . . 4 ((𝑃 ∈ (𝐵m 𝐴) ∧ 𝑋 ∈ (𝐵m 𝐴)) → (𝑃𝑇𝑋 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))))
62, 3, 5syl2anc 582 . . 3 (𝜑 → (𝑃𝑇𝑋 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))))
71, 6mpbid 231 . 2 (𝜑 → ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))
8 wemaplem3.xq . . 3 (𝜑𝑋𝑇𝑄)
9 wemaplem2.q . . . 4 (𝜑𝑄 ∈ (𝐵m 𝐴))
104wemaplem1 9569 . . . 4 ((𝑋 ∈ (𝐵m 𝐴) ∧ 𝑄 ∈ (𝐵m 𝐴)) → (𝑋𝑇𝑄 ↔ ∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))))
113, 9, 10syl2anc 582 . . 3 (𝜑 → (𝑋𝑇𝑄 ↔ ∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))))
128, 11mpbid 231 . 2 (𝜑 → ∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))
132ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑃 ∈ (𝐵m 𝐴))
143ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑋 ∈ (𝐵m 𝐴))
159ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑄 ∈ (𝐵m 𝐴))
16 wemaplem2.r . . . . . 6 (𝜑𝑅 Or 𝐴)
1716ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑅 Or 𝐴)
18 wemaplem2.s . . . . . 6 (𝜑𝑆 Po 𝐵)
1918ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑆 Po 𝐵)
20 simplrl 775 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑎𝐴)
21 simp2rl 1239 . . . . . 6 ((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → (𝑃𝑎)𝑆(𝑋𝑎))
22213expa 1115 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → (𝑃𝑎)𝑆(𝑋𝑎))
23 simprr 771 . . . . . 6 ((𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))) → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))
2423ad2antlr 725 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))
25 simprl 769 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑏𝐴)
26 simprrl 779 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → (𝑋𝑏)𝑆(𝑄𝑏))
27 simprrr 780 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))
284, 13, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27wemaplem2 9570 . . . 4 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑃𝑇𝑄)
2928rexlimdvaa 3146 . . 3 ((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) → (∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → 𝑃𝑇𝑄))
3029rexlimdvaa 3146 . 2 (𝜑 → (∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))) → (∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → 𝑃𝑇𝑄)))
317, 12, 30mp2d 49 1 (𝜑𝑃𝑇𝑄)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3051  wrex 3060   class class class wbr 5143  {copab 5205   Po wpo 5582   Or wor 5583  cfv 6543  (class class class)co 7416  m cmap 8843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-map 8845
This theorem is referenced by:  wemappo  9572
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