Theorem rp-tfslim 43343

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 → ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rp-tfslim

Step	Hyp	Ref	Expression
1		fvex 6829	. . 3 ⊢ (𝐴‘𝑥) ∈ V
2	1	dfiun3 5905	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥))
3		dffn5 6874	. . . . 5 ⊢ (𝐴 Fn 𝐵 ↔ 𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
4	3	biimpi 216	. . . 4 ⊢ (𝐴 Fn 𝐵 → 𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
5	4	rneqd 5874	. . 3 ⊢ (𝐴 Fn 𝐵 → ran 𝐴 = ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
6	5	unieqd 4869	. 2 ⊢ (𝐴 Fn 𝐵 → ∪ ran 𝐴 = ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
7	2, 6	eqtr4id 2783	1 ⊢ (𝐴 Fn 𝐵 → ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran 𝐴)

Syntax hints:   → wi 4   = wceq 1540  ∪ cuni 4856  ∪ ciun 4938   ↦ cmpt 5169  ran crn 5614   Fn wfn 6471  ‘cfv 6476

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pr 5367

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-fv 6484

This theorem is referenced by: (None)