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Theorem disjrnmpt 29730
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 29727 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
2 eqid 2770 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32rnmpt 5509 . . 3 ran (𝑥𝐴𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
4 disjeq1 4759 . . 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 224 1 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ ran (𝑥𝐴𝐵)𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1630  {cab 2756  wrex 3061  Disj wdisj 4752  cmpt 4861  ran crn 5250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-disj 4753  df-br 4785  df-opab 4845  df-mpt 4862  df-cnv 5257  df-dm 5259  df-rn 5260
This theorem is referenced by:  sigapildsys  30559  ldgenpisyslem1  30560  carsgclctunlem2  30715  pmeasadd  30721
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