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Theorem rp-tfslim 43366
Description: The limit of a sequence of ordinals is the union of its range. (Contributed by RP, 1-Mar-2025.)
Assertion
Ref Expression
rp-tfslim (𝐴 Fn 𝐵 𝑥𝐵 (𝐴𝑥) = ran 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rp-tfslim
StepHypRef Expression
1 fvex 6919 . . 3 (𝐴𝑥) ∈ V
21dfiun3 5980 . 2 𝑥𝐵 (𝐴𝑥) = ran (𝑥𝐵 ↦ (𝐴𝑥))
3 dffn5 6967 . . . . 5 (𝐴 Fn 𝐵𝐴 = (𝑥𝐵 ↦ (𝐴𝑥)))
43biimpi 216 . . . 4 (𝐴 Fn 𝐵𝐴 = (𝑥𝐵 ↦ (𝐴𝑥)))
54rneqd 5949 . . 3 (𝐴 Fn 𝐵 → ran 𝐴 = ran (𝑥𝐵 ↦ (𝐴𝑥)))
65unieqd 4920 . 2 (𝐴 Fn 𝐵 ran 𝐴 = ran (𝑥𝐵 ↦ (𝐴𝑥)))
72, 6eqtr4id 2796 1 (𝐴 Fn 𝐵 𝑥𝐵 (𝐴𝑥) = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540   cuni 4907   ciun 4991  cmpt 5225  ran crn 5686   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by: (None)
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