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Theorem ovmpt4g 5859
Description: Value of a function given by the maps-to notation. (This is the operation analog of fvmpt2 5470.) (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 2672 . . 3 (𝐶𝑉 → ∃𝑧 𝑧 = 𝐶)
2 moeq 2830 . . . . . . 7 ∃*𝑧 𝑧 = 𝐶
32a1i 9 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ∃*𝑧 𝑧 = 𝐶)
4 ovmpt4g.3 . . . . . . 7 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
5 df-mpo 5745 . . . . . . 7 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
64, 5eqtri 2136 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
73, 6ovidi 5855 . . . . 5 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧))
8 eqeq2 2125 . . . . 5 (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝐶))
97, 8mpbidi 150 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
109exlimdv 1773 . . 3 ((𝑥𝐴𝑦𝐵) → (∃𝑧 𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶))
111, 10syl5 32 . 2 ((𝑥𝐴𝑦𝐵) → (𝐶𝑉 → (𝑥𝐹𝑦) = 𝐶))
12113impia 1161 1 ((𝑥𝐴𝑦𝐵𝐶𝑉) → (𝑥𝐹𝑦) = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 945   = wceq 1314  wex 1451  wcel 1463  ∃*wmo 1976  (class class class)co 5740  {coprab 5741  cmpo 5742
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745
This theorem is referenced by:  ovmpos  5860  ov2gf  5861  ovmpodxf  5862  ovmpodf  5868  ofmres  6000  fnmpoovd  6078  mapxpen  6708  cnmpt21  12355  cnmpt2t  12357  cnmptcom  12362
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