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Theorem relmptopab 7649
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 7000 . . 3 (∀𝑥𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹𝐵) ⊆ (V × V))
3 relopab 5814 . . . . 5 Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4 df-rel 5673 . . . . 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 3059 . 2 (𝐹𝐵) ⊆ (V × V)
8 df-rel 5673 . 2 (Rel (𝐹𝐵) ↔ (𝐹𝐵) ⊆ (V × V))
97, 8mpbir 230 1 Rel (𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3466  wss 3940  {copab 5200  cmpt 5221   × cxp 5664  Rel wrel 5671  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fv 6541
This theorem is referenced by:  reldvdsr  20251  lmrel  23055  phtpcrel  24840  ulmrel  26230  ercgrg  28203  relwlk  29318  reltrls  29386  relpths  29412  releupth  29887  acycgr0v  34594  prclisacycgr  34597
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