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Theorem ovmpt4g 6736
Description: Value of a function given by the "maps to" notation. (This is the operation analogue of fvmpt2 6248.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypothesis
Ref Expression
ovmpt4g.3 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
ovmpt4g ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ovmpt4g
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elisset 3201 . . 3 (𝐶𝑉 → ∃𝑧 𝑧 = 𝐶)
2 moeq 3364 . . . . . . 7 ∃*𝑧 𝑧 = 𝐶
32a1i 11 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ∃*𝑧 𝑧 = 𝐶)
4 ovmpt4g.3 . . . . . . 7 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
5 df-mpt2 6609 . . . . . . 7 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
64, 5eqtri 2643 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
73, 6ovidi 6732 . . . . 5 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧))
8 eqeq2 2632 . . . . 5 (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝐶))
97, 8mpbidi 231 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
109exlimdv 1858 . . 3 ((𝑥𝐴𝑦𝐵) → (∃𝑧 𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
111, 10syl5 34 . 2 ((𝑥𝐴𝑦𝐵) → (𝐶𝑉 → (𝑥𝐹𝑦) = 𝐶))
12113impia 1258 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  ∃*wmo 2470  (class class class)co 6604  {coprab 6605  cmpt2 6606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609
This theorem is referenced by:  ovmpt2s  6737  ov2gf  6738  ovmpt2dxf  6739  ovmpt2df  6745  ofmres  7109  fnmpt2ovd  7197  mapxpen  8070  pwfseqlem2  9425  pwfseqlem3  9426  fullfunc  16487  fthfunc  16488  prfcl  16764  curf1cl  16789  curfcl  16793  hofcl  16820  gsum2d2lem  18293  gsum2d2  18294  gsumcom2  18295  dprdval  18323  dprd2d2  18364  cnmpt21  21384  cnmpt2t  21386  cnmptcom  21391  cnmpt2k  21401  xkocnv  21527  madjusmdetlem1  29672  madjusmdetlem3  29674  finxpreclem5  32861  sdclem2  33167  smflimlem1  40283  smflimlem2  40284  aovmpt4g  40582  ovmpt2rdxf  41402
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