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 Description: Elementhood in the domain of the adjoint function. (Contributed by Mario Carneiro, 11-Sep-2015.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
Assertion
Ref Expression
adjeu (𝑇: ℋ⟶ ℋ → (𝑇 ∈ dom adj ↔ ∃!𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
Distinct variable group:   𝑥,𝑢,𝑦,𝑇

Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 28165 . . . 4 ℋ ∈ V
2 fex2 7286 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V ∧ ℋ ∈ V) → 𝑇 ∈ V)
31, 1, 2mp3an23 1565 . . 3 (𝑇: ℋ⟶ ℋ → 𝑇 ∈ V)
4 feq1 6187 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑡: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
5 fveq1 6351 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
65oveq2d 6829 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑥 ·ih (𝑡𝑦)) = (𝑥 ·ih (𝑇𝑦)))
76eqeq1d 2762 . . . . . . . . . 10 (𝑡 = 𝑇 → ((𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
872ralbidv 3127 . . . . . . . . 9 (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
94, 83anbi13d 1550 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
10 3anass 1081 . . . . . . . 8 ((𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
119, 10syl6bb 276 . . . . . . 7 (𝑡 = 𝑇 → ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))))
1211exbidv 1999 . . . . . 6 (𝑡 = 𝑇 → (∃𝑢(𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ ∃𝑢(𝑇: ℋ⟶ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))))
13 19.42v 2030 . . . . . 6 (∃𝑢(𝑇: ℋ⟶ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))) ↔ (𝑇: ℋ⟶ ℋ ∧ ∃𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
1412, 13syl6bb 276 . . . . 5 (𝑡 = 𝑇 → (∃𝑢(𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ ∃𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))))
15 dfadj2 29053 . . . . . . 7 adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))}
1615dmeqi 5480 . . . . . 6 dom adj = dom {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))}
17 dmopab 5490 . . . . . 6 dom {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))} = {𝑡 ∣ ∃𝑢(𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))}
1816, 17eqtri 2782 . . . . 5 dom adj = {𝑡 ∣ ∃𝑢(𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))}
1914, 18elab2g 3493 . . . 4 (𝑇 ∈ V → (𝑇 ∈ dom adj ↔ (𝑇: ℋ⟶ ℋ ∧ ∃𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))))
2019baibd 986 . . 3 ((𝑇 ∈ V ∧ 𝑇: ℋ⟶ ℋ) → (𝑇 ∈ dom adj ↔ ∃𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
213, 20mpancom 706 . 2 (𝑇: ℋ⟶ ℋ → (𝑇 ∈ dom adj ↔ ∃𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
22 df-reu 3057 . . 3 (∃!𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ ∃!𝑢(𝑢 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
231, 1elmap 8052 . . . . 5 (𝑢 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑢: ℋ⟶ ℋ)
2423anbi1i 733 . . . 4 ((𝑢 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
2524eubii 2629 . . 3 (∃!𝑢(𝑢 ∈ ( ℋ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ ∃!𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
26 adjmo 29000 . . . 4 ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))
27 eu5 2633 . . . 4 (∃!𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (∃𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ∧ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))))
2826, 27mpbiran2 992 . . 3 (∃!𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ ∃𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
2922, 25, 283bitri 286 . 2 (∃!𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ ∃𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
3021, 29syl6bbr 278 1 (𝑇: ℋ⟶ ℋ → (𝑇 ∈ dom adj ↔ ∃!𝑢 ∈ ( ℋ ↑𝑚 ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632  ∃wex 1853   ∈ wcel 2139  ∃!weu 2607  ∃*wmo 2608  {cab 2746  ∀wral 3050  ∃!wreu 3052  Vcvv 3340  {copab 4864  dom cdm 5266  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ↑𝑚 cmap 8023   ℋchil 28085   ·ih csp 28088  adjℎcado 28121 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-hilex 28165  ax-hfvadd 28166  ax-hvcom 28167  ax-hvass 28168  ax-hv0cl 28169  ax-hvaddid 28170  ax-hfvmul 28171  ax-hvmulid 28172  ax-hvdistr2 28175  ax-hvmul0 28176  ax-hfi 28245  ax-his1 28248  ax-his2 28249  ax-his3 28250  ax-his4 28251 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-2 11271  df-cj 14038  df-re 14039  df-im 14040  df-hvsub 28137  df-adjh 29017 This theorem is referenced by:  adjbdln  29251
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