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

Theorem relmptopab 7607
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 6964 . . 3 (∀𝑥𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹𝐵) ⊆ (V × V))
3 relopab 5784 . . . . 5 Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4 df-rel 5644 . . . . 5 (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
53, 4mpbi 229 . . . 4 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
65a1i 11 . . 3 (𝑥𝐴 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
72, 6mprg 3067 . 2 (𝐹𝐵) ⊆ (V × V)
8 df-rel 5644 . 2 (Rel (𝐹𝐵) ↔ (𝐹𝐵) ⊆ (V × V))
97, 8mpbir 230 1 Rel (𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3447  wss 3914  {copab 5171  cmpt 5192   × cxp 5635  Rel wrel 5642  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508
This theorem is referenced by:  reldvdsr  20081  lmrel  22604  phtpcrel  24379  ulmrel  25760  ercgrg  27508  relwlk  28623  reltrls  28691  relpths  28717  releupth  29192  acycgr0v  33806  prclisacycgr  33809
  Copyright terms: Public domain W3C validator