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Theorem aovmpt4g 41881
Description: Value of a function given by the maps-to notation, analogous to ovmpt4g 6980. (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)
21dmmpt2g 7443 . . . . . 6 (𝐶𝑉 → dom 𝐹 = (𝐴 × 𝐵))
3 opelxpi 5313 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
4 eleq2 2832 . . . . . . 7 (dom 𝐹 = (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
53, 4syl5ibr 237 . . . . . 6 (dom 𝐹 = (𝐴 × 𝐵) → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
62, 5syl 17 . . . . 5 (𝐶𝑉 → ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
76impcom 396 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ 𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
873impa 1136 . . 3 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
91mpt2fun 6959 . . . 4 Fun 𝐹
10 funres 6109 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}))
119, 10ax-mp 5 . . 3 Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})
12 df-dfat 41799 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})))
13 aovfundmoveq 41861 . . . 4 (𝐹 defAt ⟨𝑥, 𝑦⟩ → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
1412, 13sylbir 226 . . 3 ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
158, 11, 14sylancl 580 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦))
161ovmpt4g 6980 . 2 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
1715, 16eqtrd 2798 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → ((𝑥𝐹𝑦)) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  {csn 4333  cop 4339   × cxp 5274  dom cdm 5276  cres 5278  Fun wfun 6061  (class class class)co 6841  cmpt2 6843   defAt wdfat 41796   ((caov 41798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-fv 6075  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-1st 7365  df-2nd 7366  df-aiota 41760  df-dfat 41799  df-afv 41800  df-aov 41801
This theorem is referenced by: (None)
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