Theorem wemappo 9502

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 8822	. . . . . . . 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 7056	. . . . . 6 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑏 ∈ 𝐴) → (𝑎‘𝑏) ∈ 𝐵)
5		poirr 5558	. . . . . 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 9499	. . . 4 ⊢ ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏 ∈ 𝐴 ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐)))))
11	10	el2v 3454	. . 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 9501	. . 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 5555	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 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   class class class wbr 5107  {copab 5169   Po wpo 5544   Or wor 5545  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801

This theorem is referenced by:  wemapsolem  9503