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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 40017 | . 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (〈𝑋, 𝑌〉 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛))) |
3 | df-br 5060 | . 2 ⊢ (𝑋(𝐶‘𝑅)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝐶‘𝑅)) | |
4 | df-br 5060 | . . 3 ⊢ (𝑋(𝑅 ↑ 𝑛)𝑌 ↔ 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛)) | |
5 | 4 | rexbii 3247 | . 2 ⊢ (∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌 ↔ ∃𝑛 ∈ 𝑁 〈𝑋, 𝑌〉 ∈ (𝑅 ↑ 𝑛)) |
6 | 2, 3, 5 | 3bitr4g 316 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 Vcvv 3495 〈cop 4567 ∪ ciun 4912 class class class wbr 5059 ↦ cmpt 5139 ‘cfv 6350 (class class class)co 7150 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 |
This theorem is referenced by: brmptiunrelexpd 40021 brtrclrec 40034 brrtrclrec 40035 briunov2uz 40036 |
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