ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fmpt2d GIF version

Theorem fmpt2d 5354
Description: Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)
Hypotheses
Ref Expression
fmpt2d.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
fmpt2d.1 (𝜑𝐹 = (𝑥𝐴𝐵))
fmpt2d.3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)
Assertion
Ref Expression
fmpt2d (𝜑𝐹:𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐶   𝑦,𝐹   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem fmpt2d
StepHypRef Expression
1 fmpt2d.2 . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
21ralrimiva 2409 . . . 4 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
3 eqid 2056 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 5052 . . . 4 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 14 . . 3 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 fmpt2d.1 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 5016 . . 3 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 160 . 2 (𝜑𝐹 Fn 𝐴)
9 fmpt2d.3 . . 3 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐶)
109ralrimiva 2409 . 2 (𝜑 → ∀𝑦𝐴 (𝐹𝑦) ∈ 𝐶)
11 ffnfv 5350 . 2 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) ∈ 𝐶))
128, 10, 11sylanbrc 402 1 (𝜑𝐹:𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  wral 2323  cmpt 3845   Fn wfn 4924  wf 4925  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator