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Theorem aovmpt4g 47116
Description: Value of a function given by the maps-to notation, analogous to ovmpt4g 7597. (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 8115 . . . . . 6 (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
3 opelxpi 5737 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
4 eleq2 2833 . . . . . . 7 (dom 𝐹 = (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
53, 4imbitrrid 246 . . . . . 6 (dom 𝐹 = (𝐴 × 𝐵) → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
62, 5syl 17 . . . . 5 (𝐶𝑉 → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
76impcom 407 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
873impa 1110 . . 3 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
91mpofun 7574 . . . 4 Fun 𝐹
10 funres 6620 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}))
119, 10ax-mp 5 . . 3 Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})
12 df-dfat 47034 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})))
13 aovfundmoveq 47096 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
1412, 13sylbir 235 . . 3 ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
158, 11, 14sylancl 585 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
161ovmpt4g 7597 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
1715, 16eqtrd 2780 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2108  {csn 4648  cop 4654   × cxp 5698  dom cdm 5700  cres 5702  Fun wfun 6567  (class class class)co 7448  cmpo 7450   defAt wdfat 47031   ((caov 47033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-aiota 47000  df-dfat 47034  df-afv 47035  df-aov 47036
This theorem is referenced by: (None)
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