Theorem wemappo 9444

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.) (Revised by AV, 21-Jul-2024.)

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	⊢ ((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)

Proof of Theorem wemappo

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpllr 775	. . . . . 6 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑏 ∈ 𝐴) → 𝑆 Po 𝐵)
2		elmapi 8781	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐵 ↑m 𝐴) → 𝑎:𝐴⟶𝐵)
3	2	adantl 481	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) → 𝑎:𝐴⟶𝐵)
4	3	ffvelcdmda 7025	. . . . . 6 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑏 ∈ 𝐴) → (𝑎‘𝑏) ∈ 𝐵)
5		poirr 5541	. . . . . 6 ⊢ ((𝑆 Po 𝐵 ∧ (𝑎‘𝑏) ∈ 𝐵) → ¬ (𝑎‘𝑏)𝑆(𝑎‘𝑏))
6	1, 4, 5	syl2anc 584	. . . . 5 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑏 ∈ 𝐴) → ¬ (𝑎‘𝑏)𝑆(𝑎‘𝑏))
7	6	intnanrd 489	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑏 ∈ 𝐴) → ¬ ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐))))
8	7	nrexdv 3128	. . 3 ⊢ (((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		wemapso.t	. . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
10	9	wemaplem1 9441	. . . 4 ⊢ ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏 ∈ 𝐴 ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐)))))
11	10	el2v 3444	. . 3 ⊢ (𝑎𝑇𝑎 ↔ ∃𝑏 ∈ 𝐴 ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐))))
12	8, 11	sylnibr 329	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) → ¬ 𝑎𝑇𝑎)
13		simplr1 1216	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑎 ∈ (𝐵 ↑m 𝐴))
14		simplr2 1217	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑏 ∈ (𝐵 ↑m 𝐴))
15		simplr3 1218	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑐 ∈ (𝐵 ↑m 𝐴))
16		simplll 774	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑅 Or 𝐴)
17		simpllr 775	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑆 Po 𝐵)
18		simprl 770	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑎𝑇𝑏)
19		simprr 772	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑏𝑇𝑐)
20	9, 13, 14, 15, 16, 17, 18, 19	wemaplem3 9443	. . 3 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑎𝑇𝑐)
21	20	ex 412	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) → ((𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐) → 𝑎𝑇𝑐))
22	12, 21	ispod 5538	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑m 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  Vcvv 3437   class class class wbr 5095  {copab 5157   Po wpo 5527   Or wor 5528  ⟶wf 6484  ‘cfv 6488  (class class class)co 7354   ↑_m cmap 8758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-po 5529  df-so 5530  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-map 8760

This theorem is referenced by:  wemapsolem  9445