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Theorem wemapso 9495
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 3970 . 2 (𝐵m 𝐴) ⊆ (𝐵m 𝐴)
3 weso 5628 . . 3 (𝑅 We 𝐴𝑅 Or 𝐴)
43adantr 482 . 2 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑅 Or 𝐴)
5 simpr 486 . 2 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑆 Or 𝐵)
6 vex 3451 . . . . . 6 𝑎 ∈ V
76difexi 5289 . . . . 5 (𝑎𝑏) ∈ V
87dmex 7852 . . . 4 dom (𝑎𝑏) ∈ V
98a1i 11 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ∈ V)
10 wefr 5627 . . . 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 4095 . . . . 5 (𝑎𝑏) ⊆ 𝑎
13 dmss 5862 . . . . 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 8793 . . . . 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 6692 . . 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 6673 . . . . . 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 8793 . . . . . . . 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 6673 . . . . . 6 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → 𝑏 Fn 𝐴)
25 fndmdifeq0 6998 . . . . . 6 ((𝑎 Fn 𝐴𝑏 Fn 𝐴) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2620, 24, 25syl2anc 585 . . . . 5 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) = ∅ ↔ 𝑎 = 𝑏))
2726necon3bid 2985 . . . 4 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → (dom (𝑎𝑏) ≠ ∅ ↔ 𝑎𝑏))
2819, 27mpbird 257 . . 3 (((𝑅 We 𝐴𝑆 Or 𝐵) ∧ ((𝑎 ∈ (𝐵m 𝐴) ∧ 𝑏 ∈ (𝐵m 𝐴)) ∧ 𝑎𝑏)) → dom (𝑎𝑏) ≠ ∅)
29 fri 5597 . . 3 (((dom (𝑎𝑏) ∈ V ∧ 𝑅 Fr 𝐴) ∧ (dom (𝑎𝑏) ⊆ 𝐴 ∧ dom (𝑎𝑏) ≠ ∅)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)
309, 11, 18, 28, 29syl22anc 838 . 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 9494 1 ((𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  Vcvv 3447  cdif 3911  wss 3914  c0 4286   class class class wbr 5109  {copab 5171   Or wor 5548   Fr wfr 5589   We wwe 5591  dom cdm 5637   Fn wfn 6495  wf 6496  cfv 6500  (class class class)co 7361  m cmap 8771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773
This theorem is referenced by:  opsrtoslem2  21486  wepwso  41417
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