Theorem ov2ssiunov2 44050

Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14992 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 1139	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → 𝑀 ∈ 𝑁)
2		simpr 484	. . . . . 6 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
3	2	oveq2d 7384	. . . . 5 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑅 ↑ 𝑛) = (𝑅 ↑ 𝑀))
4	3	eleq2d 2823	. . . 4 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 ↑ 𝑛) ↔ 𝑥 ∈ (𝑅 ↑ 𝑀)))
5	1, 4	rspcedv 3571	. . 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 44029	. . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛)))
8	7	biimprd 248	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛) → 𝑥 ∈ (𝐶‘𝑅)))
9	8	3adant3 1133	. . 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 3941	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  ∪ ciun 4948   ↦ cmpt 5181  ‘cfv 6500  (class class class)co 7368

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371

This theorem is referenced by:  dftrcl3  44070  dfrtrcl3  44083