Theorem ov2ssiunov2 40819

Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14477 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)

Hypothesis

Ref	Expression
ov2ssiunov2.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))

Assertion

Ref	Expression
ov2ssiunov2	⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅))

Distinct variable groups:   𝑛,𝑟,𝐶,𝑁, ↑   𝑛,𝑀   𝑅,𝑟,𝑛   𝑈,𝑛   𝑛,𝑉

Allowed substitution hints:   𝑈(𝑟)   𝑀(𝑟)   𝑉(𝑟)

Proof of Theorem ov2ssiunov2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1135	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → 𝑀 ∈ 𝑁)
2		simpr 488	. . . . . 6 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
3	2	oveq2d 7172	. . . . 5 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑅 ↑ 𝑛) = (𝑅 ↑ 𝑀))
4	3	eleq2d 2837	. . . 4 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 ↑ 𝑛) ↔ 𝑥 ∈ (𝑅 ↑ 𝑀)))
5	1, 4	rspcedv 3536	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑥 ∈ (𝑅 ↑ 𝑀) → ∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛)))
6		ov2ssiunov2.def	. . . . . 6 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
7	6	eliunov2 40798	. . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛)))
8	7	biimprd 251	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛) → 𝑥 ∈ (𝐶‘𝑅)))
9	8	3adant3 1129	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛) → 𝑥 ∈ (𝐶‘𝑅)))
10	5, 9	syld 47	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑥 ∈ (𝑅 ↑ 𝑀) → 𝑥 ∈ (𝐶‘𝑅)))
11	10	ssrdv 3900	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  ∪ ciun 4886   ↦ cmpt 5116  ‘cfv 6340  (class class class)co 7156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159

This theorem is referenced by:  dftrcl3  40839  dfrtrcl3  40852