Theorem rp-tfslim 44108

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 6894	. . 3 ⊢ (𝐴‘𝑥) ∈ V
2	1	dfiun3 5959	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥))
3		dffn5 6939	. . . . 5 ⊢ (𝐴 Fn 𝐵 ↔ 𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
4	3	biimpi 219	. . . 4 ⊢ (𝐴 Fn 𝐵 → 𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
5	4	rneqd 5927	. . 3 ⊢ (𝐴 Fn 𝐵 → ran 𝐴 = ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
6	5	unieqd 4884	. 2 ⊢ (𝐴 Fn 𝐵 → ∪ ran 𝐴 = ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
7	2, 6	eqtr4id 2816	1 ⊢ (𝐴 Fn 𝐵 → ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5191  ran crn 5661   Fn wfn 6531  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is used by: (None)