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Theorem disjrnmpt 29382
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 29379 . 2 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
2 eqid 2621 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32rnmpt 5369 . . 3 ran (𝑥𝐴𝐵) = {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}
4 disjeq1 4625 . . 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 1482  {cab 2607  wrex 2912  Disj wdisj 4618  cmpt 4727  ran crn 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-disj 4619  df-br 4652  df-opab 4711  df-mpt 4728  df-cnv 5120  df-dm 5122  df-rn 5123
This theorem is referenced by:  sigapildsys  30210  ldgenpisyslem1  30211  carsgclctunlem2  30366  pmeasadd  30372
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