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Theorem wemapso 9002
 Description: Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 21-Jul-2024.)
Hypothesis
Ref Expression
wemapso.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
Assertion
Ref Expression
wemapso ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (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)

Proof of Theorem wemapso
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wemapso.t . 2 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}
2 ssid 3937 . 2 (𝐵m 𝐴) ⊆ (𝐵m 𝐴)
3 weso 5511 . . 3 (𝑅 We 𝐴𝑅 Or 𝐴)
43adantr 484 . 2 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑅 Or 𝐴)
5 simpr 488 . 2 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑆 Or 𝐵)
6 vex 3444 . . . . . 6 𝑎 ∈ V
76difexi 5197 . . . . 5 (𝑎𝑏) ∈ V
87dmex 7601 . . . 4 dom (𝑎𝑏) ∈ V
98a1i 11 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ∈ V)
10 wefr 5510 . . . 4 (𝑅 We 𝐴𝑅 Fr 𝐴)
1110ad2antrr 725 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑅 Fr 𝐴)
12 difss 4059 . . . . 5 (𝑎𝑏) ⊆ 𝑎
13 dmss 5736 . . . . 5 ((𝑎𝑏) ⊆ 𝑎 → dom (𝑎𝑏) ⊆ dom 𝑎)
1412, 13ax-mp 5 . . . 4 dom (𝑎𝑏) ⊆ dom 𝑎
15 simprll 778 . . . . 5 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎 ∈ (𝐵m 𝐴))
16 elmapi 8414 . . . . 5 (𝑎 ∈ (𝐵m 𝐴) → 𝑎:𝐴𝐵)
1715, 16syl 17 . . . 4 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎:𝐴𝐵)
1814, 17fssdm 6505 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ⊆ 𝐴)
19 simprr 772 . . . 4 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎𝑏)
2017ffnd 6489 . . . . . 6 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑎 Fn 𝐴)
21 simprlr 779 . . . . . . . 8 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏 ∈ (𝐵m 𝐴))
22 elmapi 8414 . . . . . . . 8 (𝑏 ∈ (𝐵m 𝐴) → 𝑏:𝐴𝐵)
2321, 22syl 17 . . . . . . 7 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏:𝐴𝐵)
2423ffnd 6489 . . . . . 6 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏 Fn 𝐴)
25 fndmdifeq0 6792 . . . . . 6 ((𝑎 Fn 𝐴𝑏 Fn 𝐴) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2620, 24, 25syl2anc 587 . . . . 5 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2726necon3bid 3031 . . . 4 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) ≠ ∅ ↔ 𝑎𝑏))
2819, 27mpbird 260 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ≠ ∅)
29 fri 5482 . . 3 (((dom (𝑎𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎𝑏) ⊆ 𝐴 ∧ dom (𝑎𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
309, 11, 18, 28, 29syl22anc 837 . 2 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
311, 2, 4, 5, 30wemapsolem 9001 1 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵m 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243   class class class wbr 5031  {copab 5093   Or wor 5438   Fr wfr 5476   We wwe 5478  dom cdm 5520   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136   ↑m cmap 8392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-map 8394 This theorem is referenced by:  opsrtoslem2  20732  wepwso  40030
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