Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjrnmpt Structured version   Visualization version   GIF version

Theorem disjrnmpt 32598
Description: Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)
Assertion
Ref Expression
disjrnmpt (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjrnmpt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 disjabrex 32595 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
2 eqid 2737 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32rnmpt 5968 . . 3 ran (𝑥𝐴𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
4 disjeq1 5117 . . 3 (ran (𝑥𝐴𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} → (Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦))
53, 4ax-mp 5 . 2 (Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
61, 5sylibr 234 1 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  {cab 2714  wrex 3070  Disj wdisj 5110  cmpt 5225  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  fnpreimac  32681  sigapildsys  34163  ldgenpisyslem1  34164  carsgclctunlem2  34321  pmeasadd  34327
  Copyright terms: Public domain W3C validator