Theorem wemappo 9580

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 788	. . . . . 6 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑏 ∈ 𝐴) → 𝑆 Po 𝐵)
2		elmapi 8907	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐵 ↑m 𝐴) → 𝑎:𝐴⟶𝐵)
3	2	adantl 487	. . . . . . 7 ⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) → 𝑎:𝐴⟶𝐵)
4	3	ffvelcdmda 7116	. . . . . 6 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑏 ∈ 𝐴) → (𝑎‘𝑏) ∈ 𝐵)
5		poirr 5578	. . . . . 6 ⊢ ((𝑆 Po 𝐵 ∧ (𝑎‘𝑏) ∈ 𝐵) → ¬ (𝑎‘𝑏)𝑆(𝑎‘𝑏))
6	1, 4, 5	syl2anc 596	. . . . 5 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) ∧ 𝑏 ∈ 𝐴) → ¬ (𝑎‘𝑏)𝑆(𝑎‘𝑏))
7	6	intnanrd 495	. . . 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 3161	. . 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 9577	. . . 4 ⊢ ((𝑎 ∈ V ∧ 𝑎 ∈ V) → (𝑎𝑇𝑎 ↔ ∃𝑏 ∈ 𝐴 ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐)))))
11	10	el2v 3461	. . 3 ⊢ (𝑎𝑇𝑎 ↔ ∃𝑏 ∈ 𝐴 ((𝑎‘𝑏)𝑆(𝑎‘𝑏) ∧ ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑎‘𝑐) = (𝑎‘𝑐))))
12	8, 11	sylnibr 332	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ 𝑎 ∈ (𝐵 ↑m 𝐴)) → ¬ 𝑎𝑇𝑎)
13		simplr1 1234	. . . 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 1235	. . . 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 1236	. . . 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 787	. . . 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 788	. . . 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 783	. . . 4 ⊢ ((((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝑎 ∈ (𝐵 ↑m 𝐴) ∧ 𝑏 ∈ (𝐵 ↑m 𝐴) ∧ 𝑐 ∈ (𝐵 ↑m 𝐴))) ∧ (𝑎𝑇𝑏 ∧ 𝑏𝑇𝑐)) → 𝑎𝑇𝑏)
19		simprr 785	. . . 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 9579	. . 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 418	. 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 5575	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑m 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   class class class wbr 5108  {copab 5173   Po wpo 5564   Or wor 5565  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-po 5566  df-so 5567  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887

This theorem is used by:  wemapsolem  9581