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Theorem colleq12d 44249
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 6079 . . . 4 (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥}))
43scotteqd 44233 . . 3 (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥}))
51, 4iuneq12d 5026 . 2 (𝜑 𝑥𝐴 Scott (𝐹 “ {𝑥}) = 𝑥𝐵 Scott (𝐺 “ {𝑥}))
6 df-coll 44247 . 2 (𝐹 Coll 𝐴) = 𝑥𝐴 Scott (𝐹 “ {𝑥})
7 df-coll 44247 . 2 (𝐺 Coll 𝐵) = 𝑥𝐵 Scott (𝐺 “ {𝑥})
85, 6, 73eqtr4g 2800 1 (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  {csn 4631   ciun 4996  cima 5692  Scott cscott 44231   Coll ccoll 44246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-scott 44232  df-coll 44247
This theorem is referenced by:  colleq1  44250  colleq2  44251
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