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Theorem rp-tfslim 43342
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 5982 . 2 𝑥𝐵 (𝐴𝑥) = ran (𝑥𝐵 ↦ (𝐴𝑥))
3 dffn5 6966 . . . . 5 (𝐴 Fn 𝐵𝐴 = (𝑥𝐵 ↦ (𝐴𝑥)))
43biimpi 216 . . . 4 (𝐴 Fn 𝐵𝐴 = (𝑥𝐵 ↦ (𝐴𝑥)))
54rneqd 5951 . . 3 (𝐴 Fn 𝐵 → ran 𝐴 = ran (𝑥𝐵 ↦ (𝐴𝑥)))
65unieqd 4924 . 2 (𝐴 Fn 𝐵 ran 𝐴 = ran (𝑥𝐵 ↦ (𝐴𝑥)))
72, 6eqtr4id 2793 1 (𝐴 Fn 𝐵 𝑥𝐵 (𝐴𝑥) = ran 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536   cuni 4911   ciun 4995  cmpt 5230  ran crn 5689   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by: (None)
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