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Theorem disjrnmpt 30924
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 30921 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
2 eqid 2738 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32rnmpt 5864 . . 3 ran (𝑥𝐴𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
4 disjeq1 5046 . . 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 233 1 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  {cab 2715  wrex 3065  Disj wdisj 5039  cmpt 5157  ran crn 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-cnv 5597  df-dm 5599  df-rn 5600
This theorem is referenced by:  fnpreimac  31008  sigapildsys  32130  ldgenpisyslem1  32131  carsgclctunlem2  32286  pmeasadd  32292
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