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Theorem relmptopab 7652
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 7003 . . 3 (∀𝑥𝐴 {⟨𝑦, 𝑧⟩ ∣ 𝜑} ⊆ (V × V) → (𝐹𝐵) ⊆ (V × V))
3 relopab 5817 . . . . 5 Rel {⟨𝑦, 𝑧⟩ ∣ 𝜑}
4 df-rel 5676 . . . . 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 3061 . 2 (𝐹𝐵) ⊆ (V × V)
8 df-rel 5676 . 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 3468  wss 3943  {copab 5203  cmpt 5224   × cxp 5667  Rel wrel 5674  cfv 6536
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544
This theorem is referenced by:  reldvdsr  20260  lmrel  23085  phtpcrel  24870  ulmrel  26265  ercgrg  28272  relwlk  29388  reltrls  29456  relpths  29482  releupth  29957  acycgr0v  34667  prclisacycgr  34670
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