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Theorem colleq12d 40881
Description: Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
colleq12d.1 (𝜑𝐹 = 𝐺)
colleq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
colleq12d (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))

Proof of Theorem colleq12d
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 colleq12d.2 . . 3 (𝜑𝐴 = 𝐵)
2 colleq12d.1 . . . . 5 (𝜑𝐹 = 𝐺)
32imaeq1d 5915 . . . 4 (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥}))
43scotteqd 40865 . . 3 (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥}))
51, 4iuneq12d 4933 . 2 (𝜑 𝑥𝐴 Scott (𝐹 “ {𝑥}) = 𝑥𝐵 Scott (𝐺 “ {𝑥}))
6 df-coll 40879 . 2 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
7 df-coll 40879 . 2 (𝐺 Coll 𝐵) = 𝑥𝐵 Scott (𝐺 “ {𝑥})
85, 6, 73eqtr4g 2884 1 (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  {csn 4550   ciun 4905  cima 5545  Scott cscott 40863   Coll ccoll 40878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-iun 4907  df-br 5053  df-opab 5115  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-scott 40864  df-coll 40879
This theorem is referenced by:  colleq1  40882  colleq2  40883
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