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Theorem aovmpt4g 44153
 Description: Value of a function given by the maps-to notation, analogous to ovmpt4g 7297. (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 7782 . . . . . 6 (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
3 opelxpi 5564 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
4 eleq2 2840 . . . . . . 7 (dom 𝐹 = (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
53, 4syl5ibr 249 . . . . . 6 (dom 𝐹 = (𝐴 × 𝐵) → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
62, 5syl 17 . . . . 5 (𝐶𝑉 → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
76impcom 411 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
873impa 1107 . . 3 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
91mpofun 7275 . . . 4 Fun 𝐹
10 funres 6381 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}))
119, 10ax-mp 5 . . 3 Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})
12 df-dfat 44071 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})))
13 aovfundmoveq 44133 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
1412, 13sylbir 238 . . 3 ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
158, 11, 14sylancl 589 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
161ovmpt4g 7297 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
1715, 16eqtrd 2793 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {csn 4525  ⟨cop 4531   × cxp 5525  dom cdm 5527   ↾ cres 5529  Fun wfun 6333  (class class class)co 7155   ∈ cmpo 7157   defAt wdfat 44068   ((caov 44070 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 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-aiota 44036  df-dfat 44071  df-afv 44072  df-aov 44073 This theorem is referenced by: (None)
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