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Theorem aovmpt4g 45507
Description: Value of a function given by the maps-to notation, analogous to ovmpt4g 7507. (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 8012 . . . . . 6 (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
3 opelxpi 5675 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
4 eleq2 2827 . . . . . . 7 (dom 𝐹 = (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
53, 4syl5ibr 246 . . . . . 6 (dom 𝐹 = (𝐴 × 𝐵) → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
62, 5syl 17 . . . . 5 (𝐶𝑉 → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
76impcom 409 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
873impa 1111 . . 3 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
91mpofun 7485 . . . 4 Fun 𝐹
10 funres 6548 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}))
119, 10ax-mp 5 . . 3 Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})
12 df-dfat 45425 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})))
13 aovfundmoveq 45487 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
1412, 13sylbir 234 . . 3 ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
158, 11, 14sylancl 587 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
161ovmpt4g 7507 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
1715, 16eqtrd 2777 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  {csn 4591  cop 4597   × cxp 5636  dom cdm 5638  cres 5640  Fun wfun 6495  (class class class)co 7362  cmpo 7364   defAt wdfat 45422   ((caov 45424
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-aiota 45391  df-dfat 45425  df-afv 45426  df-aov 45427
This theorem is referenced by: (None)
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