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Theorem relmptopab 7684
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 7027 . . 3 (∀𝑥𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹𝐵) ⊆ (V × V))
3 relopab 5833 . . . . 5 Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4 df-rel 5691 . . . . 5 (Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑} ↔ {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
53, 4mpbi 230 . . . 4 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
65a1i 11 . . 3 (𝑥𝐴 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
72, 6mprg 3066 . 2 (𝐹𝐵) ⊆ (V × V)
8 df-rel 5691 . 2 (Rel (𝐹𝐵) ↔ (𝐹𝐵) ⊆ (V × V))
97, 8mpbir 231 1 Rel (𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  Vcvv 3479  wss 3950  {copab 5204  cmpt 5224   × cxp 5682  Rel wrel 5689  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568
This theorem is referenced by:  reldvdsr  20361  lmrel  23239  phtpcrel  25026  ulmrel  26422  ercgrg  28526  relwlk  29645  reltrls  29713  relpths  29739  releupth  30219  acycgr0v  35154  prclisacycgr  35157
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