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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 40380 | . 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (〈𝑋, 𝑌〉 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛))) |
3 | df-br 5031 | . 2 ⊢ (𝑋(𝐶‘𝑅)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝐶‘𝑅)) | |
4 | df-br 5031 | . . 3 ⊢ (𝑋(𝑅 ↑ 𝑛)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛)) | |
5 | 4 | rexbii 3210 | . 2 ⊢ (∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ∃𝑛 ∈ 𝑁 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛)) |
6 | 2, 3, 5 | 3bitr4g 317 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 〈cop 4531 ∪ ciun 4881 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 |
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 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 |
This theorem is referenced by: brmptiunrelexpd 40384 brtrclrec 40397 brrtrclrec 40398 briunov2uz 40399 |
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