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Theorem ovmpt4g 7017
Description: Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 6516.) (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 3403 . . 3 (𝐶𝑉 → ∃𝑧 𝑧 = 𝐶)
2 moeq 3572 . . . . . . 7 ∃*𝑧 𝑧 = 𝐶
32a1i 11 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ∃*𝑧 𝑧 = 𝐶)
4 ovmpt4g.3 . . . . . . 7 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
5 df-mpt2 6883 . . . . . . 7 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
64, 5eqtri 2821 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
73, 6ovidi 7013 . . . . 5 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧))
8 eqeq2 2810 . . . . 5 (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝐶))
97, 8mpbidi 233 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
109exlimdv 2029 . . 3 ((𝑥𝐴𝑦𝐵) → (∃𝑧 𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
111, 10syl5 34 . 2 ((𝑥𝐴𝑦𝐵) → (𝐶𝑉 → (𝑥𝐹𝑦) = 𝐶))
12113impia 1146 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  ∃*wmo 2589  (class class class)co 6878  {coprab 6879  cmpt2 6880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883
This theorem is referenced by:  ovmpt2s  7018  ov2gf  7019  ovmpt2dxf  7020  ovmpt2df  7026  ofmres  7397  fnmpt2ovd  7488  fnmpt2ovdOLD  7489  mapxpen  8368  pwfseqlem2  9769  pwfseqlem3  9770  fullfunc  16880  fthfunc  16881  prfcl  17158  curf1cl  17183  curfcl  17187  hofcl  17214  gsum2d2lem  18687  gsum2d2  18688  gsumcom2  18689  dprdval  18718  dprd2d2  18759  cnmpt21  21803  cnmpt2t  21805  cnmptcom  21810  cnmpt2k  21820  xkocnv  21946  madjusmdetlem1  30409  madjusmdetlem3  30411  finxpreclem5  33730  sdclem2  34025  smflimlem1  41721  smflimlem2  41722  aovmpt4g  42051  ovmpt2rdxf  42912
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