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Theorem relmpt2opab 7146
 Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
relmpt2opab.1 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})
Assertion
Ref Expression
relmpt2opab Rel (𝐶𝐹𝐷)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑦,𝐵   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑧,𝑤)   𝐵(𝑥,𝑧,𝑤)   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑦,𝑧,𝑤)   𝐹(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem relmpt2opab
StepHypRef Expression
1 relopab 5169 . . . . 5 Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑}
2 df-rel 5045 . . . . 5 (Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} ↔ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V))
31, 2mpbi 219 . . . 4 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
43rgen2w 2909 . . 3 𝑥𝐴𝑦𝐵 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
5 relmpt2opab.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})
65ovmptss 7145 . . 3 (∀𝑥𝐴𝑦𝐵 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V) → (𝐶𝐹𝐷) ⊆ (V × V))
74, 6ax-mp 5 . 2 (𝐶𝐹𝐷) ⊆ (V × V)
8 df-rel 5045 . 2 (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V))
97, 8mpbir 220 1 Rel (𝐶𝐹𝐷)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  {copab 4642   × cxp 5036  Rel wrel 5043  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060 This theorem is referenced by:  brovmpt2ex  7236  relfunc  16345  releqg  17464  relwlk  26046  releupa  26491
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