Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ovmpt4g GIF version

Theorem ovmpt4g 5565
 Description: Value of a function given by the "maps to" notation. (This is the operation analog of fvmpt2 5197.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.)
Hypothesis
Ref Expression
ovmpt4g.3 𝐹 = (x A, y B𝐶)
Assertion
Ref Expression
ovmpt4g ((x A y B 𝐶 𝑉) → (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 2562 . . 3 (𝐶 𝑉z z = 𝐶)
2 moeq 2710 . . . . . . 7 ∃*z z = 𝐶
32a1i 9 . . . . . 6 ((x A y B) → ∃*z z = 𝐶)
4 ovmpt4g.3 . . . . . . 7 𝐹 = (x A, y B𝐶)
5 df-mpt2 5460 . . . . . . 7 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
64, 5eqtri 2057 . . . . . 6 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
73, 6ovidi 5561 . . . . 5 ((x A y B) → (z = 𝐶 → (x𝐹y) = z))
8 eqeq2 2046 . . . . 5 (z = 𝐶 → ((x𝐹y) = z ↔ (x𝐹y) = 𝐶))
97, 8mpbidi 140 . . . 4 ((x A y B) → (z = 𝐶 → (x𝐹y) = 𝐶))
109exlimdv 1697 . . 3 ((x A y B) → (z z = 𝐶 → (x𝐹y) = 𝐶))
111, 10syl5 28 . 2 ((x A y B) → (𝐶 𝑉 → (x𝐹y) = 𝐶))
12113impia 1100 1 ((x A y B 𝐶 𝑉) → (x𝐹y) = 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  (class class class)co 5455  {coprab 5456   ↦ cmpt2 5457 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460 This theorem is referenced by:  ovmpt2s  5566  ov2gf  5567  ovmpt2dxf  5568  ovmpt2df  5574  ofmres  5705
 Copyright terms: Public domain W3C validator