MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relmpoopab Structured version   Visualization version   GIF version

Theorem relmpoopab 8080
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 5814 . . . . 5 Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑}
2 df-rel 5676 . . . . 5 (Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} ↔ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V))
31, 2mpbi 229 . . . 4 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
43rgen2w 3060 . . 3 𝑥𝐴𝑦𝐵 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)
5 relmpoopab.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨𝑧, 𝑤⟩ ∣ 𝜑})
65ovmptss 8079 . . 3 (∀𝑥𝐴𝑦𝐵 {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V) → (𝐶𝐹𝐷) ⊆ (V × V))
74, 6ax-mp 5 . 2 (𝐶𝐹𝐷) ⊆ (V × V)
8 df-rel 5676 . 2 (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V))
97, 8mpbir 230 1 Rel (𝐶𝐹𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wral 3055  Vcvv 3468  wss 3943  {copab 5203   × cxp 5667  Rel wrel 5674  (class class class)co 7405  cmpo 7407
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  ax-un 7722
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-iun 4992  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 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975
This theorem is referenced by:  brovmpoex  8209  relfunc  17821  releqg  19102
  Copyright terms: Public domain W3C validator