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Theorem ovmpt4g 5861
Description: Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5472.) (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 2674 . . 3 (𝐶𝑉 → ∃𝑧 𝑧 = 𝐶)
2 moeq 2832 . . . . . . 7 ∃*𝑧 𝑧 = 𝐶
32a1i 9 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ∃*𝑧 𝑧 = 𝐶)
4 ovmpt4g.3 . . . . . . 7 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
5 df-mpo 5747 . . . . . . 7 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
64, 5eqtri 2138 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
73, 6ovidi 5857 . . . . 5 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧))
8 eqeq2 2127 . . . . 5 (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝐶))
97, 8mpbidi 150 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
109exlimdv 1775 . . 3 ((𝑥𝐴𝑦𝐵) → (∃𝑧 𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
111, 10syl5 32 . 2 ((𝑥𝐴𝑦𝐵) → (𝐶𝑉 → (𝑥𝐹𝑦) = 𝐶))
12113impia 1163 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wex 1453  wcel 1465  ∃*wmo 1978  (class class class)co 5742  {coprab 5743  cmpo 5744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747
This theorem is referenced by:  ovmpos  5862  ov2gf  5863  ovmpodxf  5864  ovmpodf  5870  ofmres  6002  fnmpoovd  6080  mapxpen  6710  cnmpt21  12387  cnmpt2t  12389  cnmptcom  12394
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