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Theorem wemappo 8619
Description: Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
Assertion
Ref Expression
wemappo ((𝐴𝑉𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝑅,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑦,𝑧,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wemappo
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3352 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpll3 1259 . . . . . . 7 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → 𝑆 Po 𝐵)
3 elmapi 8045 . . . . . . . . 9 (𝑎 ∈ (𝐵𝑚 𝐴) → 𝑎:𝐴𝐵)
43adantl 473 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → 𝑎:𝐴𝐵)
54ffvelrnda 6522 . . . . . . 7 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → (𝑎𝑏) ∈ 𝐵)
6 poirr 5198 . . . . . . 7 ((𝑆 Po 𝐵 ∧ (𝑎𝑏) ∈ 𝐵) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
72, 5, 6syl2anc 696 . . . . . 6 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → ¬ (𝑎𝑏)𝑆(𝑎𝑏))
87intnanrd 1001 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) ∧ 𝑏𝐴) → ¬ ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
98nrexdv 3139 . . . 4 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → ¬ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
10 vex 3343 . . . . 5 𝑎 ∈ V
11 wemapso.t . . . . . 6 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
1211wemaplem1 8616 . . . . 5 ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐)))))
1310, 10, 12mp2an 710 . . . 4 (𝑎𝑇𝑎 ↔ ∃𝑏𝐴 ((𝑎𝑏)𝑆(𝑎𝑏) ∧ ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑎𝑐) = (𝑎𝑐))))
149, 13sylnibr 318 . . 3 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵𝑚 𝐴)) → ¬ 𝑎𝑇𝑎)
15 simpll1 1255 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝐴 ∈ V)
16 simplr1 1261 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵𝑚 𝐴))
17 simplr2 1263 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵𝑚 𝐴))
18 simplr3 1265 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑐 ∈ (𝐵𝑚 𝐴))
19 simpll2 1257 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑅 Or 𝐴)
20 simpll3 1259 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑆 Po 𝐵)
21 simprl 811 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑏)
22 simprr 813 . . . . 5 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑏𝑇𝑐)
2311, 15, 16, 17, 18, 19, 20, 21, 22wemaplem3 8618 . . . 4 ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) ∧ (𝑎𝑇𝑏𝑏𝑇𝑐)) → 𝑎𝑇𝑐)
2423ex 449 . . 3 (((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵𝑚 𝐴) ∧ 𝑏 ∈ (𝐵𝑚 𝐴) ∧ 𝑐 ∈ (𝐵𝑚 𝐴))) → ((𝑎𝑇𝑏𝑏𝑇𝑐) → 𝑎𝑇𝑐))
2514, 24ispod 5195 . 2 ((𝐴 ∈ V ∧ 𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))
261, 25syl3an1 1167 1 ((𝐴𝑉𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340   class class class wbr 4804  {copab 4864   Po wpo 5185   Or wor 5186  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025
This theorem is referenced by:  wemapsolem  8620
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