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Theorem relmpoopab 7800
 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
relmpoopab.1 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})
Assertion
Ref Expression
relmpoopab 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 relmpoopab
StepHypRef Expression
1 relopabv 5668 . . . . 5 Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑}
2 df-rel 5535 . . . . 5 (Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} ↔ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V))
31, 2mpbi 233 . . . 4 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
43rgen2w 3083 . . 3 𝑥𝐴𝑦𝐵 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
5 relmpoopab.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})
65ovmptss 7799 . . 3 (∀𝑥𝐴𝑦𝐵 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V) → (𝐶𝐹𝐷) ⊆ (V × V))
74, 6ax-mp 5 . 2 (𝐶𝐹𝐷) ⊆ (V × V)
8 df-rel 5535 . 2 (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V))
97, 8mpbir 234 1 Rel (𝐶𝐹𝐷)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ∀wral 3070  Vcvv 3409   ⊆ wss 3860  {copab 5098   × cxp 5526  Rel wrel 5533  (class class class)co 7156   ∈ cmpo 7158 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700 This theorem is referenced by:  brovmpoex  7905  relfunc  17204  releqg  18407
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