Theorem wemappo 8849

Description: Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values.

Without totality on the values or least differing indices, 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⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}

Assertion

Ref	Expression
wemappo	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑𝑚 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.

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		simpll3 1205	. . . . . . 7 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑏 ∈ 𝐴) → 𝑆 Po 𝐵)
3		elmapi 8269	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) → 𝑎:𝐴⟶𝐵)
4	3	adantl 482	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑𝑚 𝐴)) → 𝑎:𝐴⟶𝐵)
5	4	ffvelrnda 6707	. . . . . . 7 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑏 ∈ 𝐴) → (𝑎‘𝑏) ∈ 𝐵)
6		poirr 5365	. . . . . . 7 ⊢ ((𝑆 Po 𝐵 ∧ (𝑎‘𝑏) ∈ 𝐵) → ¬ (𝑎‘𝑏)𝑆(𝑎‘𝑏))
7	2, 5, 6	syl2anc 584	. . . . . 6 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑏 ∈ 𝐴) → ¬ (𝑎‘𝑏)𝑆(𝑎‘𝑏))
8	7	intnanrd 490	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑𝑚 𝐴)) ∧ 𝑏 ∈ 𝐴) → ¬ ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐))))
9	8	nrexdv 3230	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑𝑚 𝐴)) → ¬ ∃𝑏 ∈ 𝐴 ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐))))
10		wemapso.t	. . . . . 6 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
11	10	wemaplem1 8846	. . . . 5 ⊢ ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏 ∈ 𝐴 ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐)))))
12	11	el2v 3439	. . . 4 ⊢ (𝑎𝑇𝑎 ↔ ∃𝑏 ∈ 𝐴 ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐))))
13	9, 12	sylnibr 330	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑𝑚 𝐴)) → ¬ 𝑎𝑇𝑎)
14		simpll1 1203	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝐴 ∈ V)
15		simplr1 1206	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵 ↑𝑚 𝐴))
16		simplr2 1207	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵 ↑𝑚 𝐴))
17		simplr3 1208	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑐 ∈ (𝐵 ↑𝑚 𝐴))
18		simpll2 1204	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑅 Or 𝐴)
19		simpll3 1205	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑆 Po 𝐵)
20		simprl 767	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑎𝑇𝑏)
21		simprr 769	. . . . 5 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑏𝑇𝑐)
22	10, 14, 15, 16, 17, 18, 19, 20, 21	wemaplem3 8848	. . . 4 ⊢ ((((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑎𝑇𝑐)
23	22	ex 413	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑏 ∈ (𝐵 ↑𝑚 𝐴) ∧ 𝑐 ∈ (𝐵 ↑𝑚 𝐴))) → ((𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐) → 𝑎𝑇𝑐))
24	13, 23	ispod 5362	. 2 ⊢ ((𝐴 ∈ V ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑𝑚 𝐴))
25	1, 24	syl3an1 1154	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑𝑚 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079  ∀wral 3103  ∃wrex 3104  Vcvv 3432   class class class wbr 4956  {copab 5018   Po wpo 5352   Or wor 5353  ⟶wf 6213  ‘cfv 6217  (class class class)co 7007   ↑_𝑚 cmap 8247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-po 5354  df-so 5355  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-1st 7536  df-2nd 7537  df-map 8249

This theorem is referenced by:  wemapsolem  8850