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Theorem colleq12d 44277
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 6076 . . . 4 (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥}))
43scotteqd 44261 . . 3 (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥}))
51, 4iuneq12d 5020 . 2 (𝜑 𝑥𝐴 Scott (𝐹 “ {𝑥}) = 𝑥𝐵 Scott (𝐺 “ {𝑥}))
6 df-coll 44275 . 2 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
7 df-coll 44275 . 2 (𝐺 Coll 𝐵) = 𝑥𝐵 Scott (𝐺 “ {𝑥})
85, 6, 73eqtr4g 2801 1 (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  {csn 4625   ciun 4990  cima 5687  Scott cscott 44259   Coll ccoll 44274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-iun 4992  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-scott 44260  df-coll 44275
This theorem is referenced by:  colleq1  44278  colleq2  44279
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