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Theorem aovmpt4g 43277
Description: Value of a function given by the maps-to notation, analogous to ovmpt4g 7286. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
aovmpt4g.3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
aovmpt4g ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem aovmpt4g
StepHypRef Expression
1 aovmpt4g.3 . . . . . . 7 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
21dmmpog 7761 . . . . . 6 (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
3 opelxpi 5585 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
4 eleq2 2898 . . . . . . 7 (dom 𝐹 = (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
53, 4syl5ibr 247 . . . . . 6 (dom 𝐹 = (𝐴 × 𝐵) → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
62, 5syl 17 . . . . 5 (𝐶𝑉 → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
76impcom 408 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
873impa 1102 . . 3 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
91mpofun 7265 . . . 4 Fun 𝐹
10 funres 6390 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}))
119, 10ax-mp 5 . . 3 Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})
12 df-dfat 43195 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})))
13 aovfundmoveq 43257 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
1412, 13sylbir 236 . . 3 ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
158, 11, 14sylancl 586 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
161ovmpt4g 7286 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
1715, 16eqtrd 2853 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  {csn 4557  cop 4563   × cxp 5546  dom cdm 5548  cres 5550  Fun wfun 6342  (class class class)co 7145  cmpo 7147   defAt wdfat 43192   ((caov 43194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-aiota 43162  df-dfat 43195  df-afv 43196  df-aov 43197
This theorem is referenced by: (None)
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