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Mirrors > Home > MPE Home > Th. List > Mathboxes > briunov2 | Structured version Visualization version GIF version |
Description: Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.) |
Ref | Expression |
---|---|
briunov2.def | ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) |
Ref | Expression |
---|---|
briunov2 | ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | briunov2.def | . . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) | |
2 | 1 | eliunov2 43669 | . 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (〈𝑋, 𝑌〉 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛))) |
3 | df-br 5149 | . 2 ⊢ (𝑋(𝐶‘𝑅)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝐶‘𝑅)) | |
4 | df-br 5149 | . . 3 ⊢ (𝑋(𝑅 ↑ 𝑛)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛)) | |
5 | 4 | rexbii 3092 | . 2 ⊢ (∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ∃𝑛 ∈ 𝑁 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛)) |
6 | 2, 3, 5 | 3bitr4g 314 | 1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 〈cop 4637 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 |
This theorem is referenced by: brmptiunrelexpd 43673 brtrclrec 43686 brrtrclrec 43687 briunov2uz 43688 |
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