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Mirrors > Home > MPE Home > Th. List > Mathboxes > ov2ssiunov2 | Structured version Visualization version GIF version |
Description: Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 13646 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.) |
Ref | Expression |
---|---|
ov2ssiunov2.def | ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) |
Ref | Expression |
---|---|
ov2ssiunov2 | ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1056 | . . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → 𝑀 ∈ 𝑁) | |
2 | simpr 476 | . . . . . 6 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀) | |
3 | 2 | oveq2d 6565 | . . . . 5 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑅 ↑ 𝑛) = (𝑅 ↑ 𝑀)) |
4 | 3 | eleq2d 2673 | . . . 4 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) ∧ 𝑛 = 𝑀) → (𝑥 ∈ (𝑅 ↑ 𝑛) ↔ 𝑥 ∈ (𝑅 ↑ 𝑀))) |
5 | 1, 4 | rspcedv 3286 | . . 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 36990 | . . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑥 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛))) |
8 | 7 | biimprd 237 | . . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛) → 𝑥 ∈ (𝐶‘𝑅))) |
9 | 8 | 3adant3 1074 | . . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (∃𝑛 ∈ 𝑁 𝑥 ∈ (𝑅 ↑ 𝑛) → 𝑥 ∈ (𝐶‘𝑅))) |
10 | 5, 9 | syld 46 | . 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑥 ∈ (𝑅 ↑ 𝑀) → 𝑥 ∈ (𝐶‘𝑅))) |
11 | 10 | ssrdv 3574 | 1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ∪ ciun 4455 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 |
This theorem is referenced by: dftrcl3 37031 dfrtrcl3 37044 |
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