Theorem rp-tfslim 44355

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 6898	. . 3 ⊢ (𝐴‘𝑥) ∈ V
2	1	dfiun3 5951	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥))
3		dffn5 6943	. . . . 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 5919	. . 3 ⊢ (𝐴 Fn 𝐵 → ran 𝐴 = ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
6	5	unieqd 4879	. 2 ⊢ (𝐴 Fn 𝐵 → ∪ ran 𝐴 = ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)))
7	2, 6	eqtr4id 2814	1 ⊢ (𝐴 Fn 𝐵 → ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185  ran crn 5651   Fn wfn 6533  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by: (None)