MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relmptopab Structured version   Visualization version   GIF version

Theorem relmptopab 7384
Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
relmptopab.1 𝐹 = (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ 𝜑})
Assertion
Ref Expression
relmptopab Rel (𝐹𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem relmptopab
StepHypRef Expression
1 relmptopab.1 . . . 4 𝐹 = (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ 𝜑})
21fvmptss 6772 . . 3 (∀𝑥𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹𝐵) ⊆ (V × V))
3 relopab 5689 . . . . 5 Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4 df-rel 5555 . . . . 5 (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
53, 4mpbi 231 . . . 4 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
65a1i 11 . . 3 (𝑥𝐴 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
72, 6mprg 3149 . 2 (𝐹𝐵) ⊆ (V × V)
8 df-rel 5555 . 2 (Rel (𝐹𝐵) ↔ (𝐹𝐵) ⊆ (V × V))
97, 8mpbir 232 1 Rel (𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  {copab 5119  cmpt 5137   × cxp 5546  Rel wrel 5553  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  reldvdsr  19323  lmrel  21766  phtpcrel  23524  ulmrel  24893  ercgrg  26230  relwlk  27334  reltrls  27403  relpths  27428  releupth  27905  acycgr0v  32292  prclisacycgr  32295
  Copyright terms: Public domain W3C validator