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Theorem wemaplem3 8609
 Description: Lemma for wemapso 8612. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Hypotheses
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
wemaplem2.a (𝜑𝐴 ∈ V)
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 8607 . . . 4 ((𝑃 ∈ (𝐵𝑚 𝐴) ∧ 𝑋 ∈ (𝐵𝑚 𝐴)) → (𝑃𝑇𝑋 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))))
62, 3, 5syl2anc 573 . . 3 (𝜑 → (𝑃𝑇𝑋 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))))
71, 6mpbid 222 . 2 (𝜑 → ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))
8 wemaplem3.xq . . 3 (𝜑𝑋𝑇𝑄)
9 wemaplem2.q . . . 4 (𝜑𝑄 ∈ (𝐵𝑚 𝐴))
104wemaplem1 8607 . . . 4 ((𝑋 ∈ (𝐵𝑚 𝐴) ∧ 𝑄 ∈ (𝐵𝑚 𝐴)) → (𝑋𝑇𝑄 ↔ ∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))))
113, 9, 10syl2anc 573 . . 3 (𝜑 → (𝑋𝑇𝑄 ↔ ∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))))
128, 11mpbid 222 . 2 (𝜑 → ∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))
13 wemaplem2.a . . . . . 6 (𝜑𝐴 ∈ V)
1413ad2antrr 705 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝐴 ∈ V)
152ad2antrr 705 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑃 ∈ (𝐵𝑚 𝐴))
163ad2antrr 705 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑋 ∈ (𝐵𝑚 𝐴))
179ad2antrr 705 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑄 ∈ (𝐵𝑚 𝐴))
18 wemaplem2.r . . . . . 6 (𝜑𝑅 Or 𝐴)
1918ad2antrr 705 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑅 Or 𝐴)
20 wemaplem2.s . . . . . 6 (𝜑𝑆 Po 𝐵)
2120ad2antrr 705 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑆 Po 𝐵)
22 simplrl 762 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑎𝐴)
23 simp2rl 1308 . . . . . 6 ((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → (𝑃𝑎)𝑆(𝑋𝑎))
24233expa 1111 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → (𝑃𝑎)𝑆(𝑋𝑎))
25 simprr 756 . . . . . 6 ((𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))) → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))
2625ad2antlr 706 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))
27 simprl 754 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑏𝐴)
28 simprrl 766 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → (𝑋𝑏)𝑆(𝑄𝑏))
29 simprrr 767 . . . . 5 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))
304, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 28, 29wemaplem2 8608 . . . 4 (((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) ∧ (𝑏𝐴 ∧ ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))))) → 𝑃𝑇𝑄)
3130rexlimdvaa 3180 . . 3 ((𝜑 ∧ (𝑎𝐴 ∧ ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))))) → (∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → 𝑃𝑇𝑄))
3231rexlimdvaa 3180 . 2 (𝜑 → (∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑋𝑎) ∧ ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐))) → (∃𝑏𝐴 ((𝑋𝑏)𝑆(𝑄𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐))) → 𝑃𝑇𝑄)))
337, 12, 32mp2d 49 1 (𝜑𝑃𝑇𝑄)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  Vcvv 3351   class class class wbr 4786  {copab 4846   Po wpo 5168   Or wor 5169  ‘cfv 6031  (class class class)co 6793   ↑𝑚 cmap 8009 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011 This theorem is referenced by:  wemappo  8610
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