Theorem ov2ssiunov2 42384

Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15000 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 486	. . . . . 6 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
3	2	oveq2d 7420	. . . . 5 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑅 ↑ 𝑛) = (𝑅 ↑ 𝑀))
4	3	eleq2d 2820	. . . 4 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 ↑ 𝑛) ↔ 𝑥 ∈ (𝑅 ↑ 𝑀)))
5	1, 4	rspcedv 3605	. . 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 42363	. . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛)))
8	7	biimprd 247	. . . 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 3987	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  Vcvv 3475   ⊆ wss 3947  ∪ ciun 4996   ↦ cmpt 5230  ‘cfv 6540  (class class class)co 7404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407

This theorem is referenced by:  dftrcl3  42404  dfrtrcl3  42417