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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnviun | Structured version Visualization version GIF version |
Description: Converse of indexed union. (Contributed by RP, 20-Jun-2020.) |
Ref | Expression |
---|---|
cnviun | ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5962 | . 2 ⊢ Rel ◡∪ 𝑥 ∈ 𝐴 𝐵 | |
2 | reliun 5684 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel ◡𝐵) | |
3 | relcnv 5962 | . . . 4 ⊢ Rel ◡𝐵 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → Rel ◡𝐵) |
5 | 2, 4 | mprgbir 3153 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
6 | vex 3498 | . . . . . 6 ⊢ 𝑦 ∈ V | |
7 | vex 3498 | . . . . . 6 ⊢ 𝑧 ∈ V | |
8 | 6, 7 | opelcnv 5747 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ◡𝐵 ↔ 〈𝑧, 𝑦〉 ∈ 𝐵) |
9 | 8 | bicomi 226 | . . . 4 ⊢ (〈𝑧, 𝑦〉 ∈ 𝐵 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐵) |
10 | 9 | rexbii 3247 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ ◡𝐵) |
11 | 6, 7 | opelcnv 5747 | . . . 4 ⊢ (〈𝑦, 𝑧〉 ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
12 | eliun 4916 | . . . 4 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵) | |
13 | 11, 12 | bitri 277 | . . 3 ⊢ (〈𝑦, 𝑧〉 ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵) |
14 | eliun 4916 | . . 3 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ ◡𝐵) | |
15 | 10, 13, 14 | 3bitr4i 305 | . 2 ⊢ (〈𝑦, 𝑧〉 ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ 〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵) |
16 | 1, 5, 15 | eqrelriiv 5658 | 1 ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ∃wrex 3139 〈cop 4567 ∪ ciun 4912 ◡ccnv 5549 Rel wrel 5555 |
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-sep 5196 ax-nul 5203 ax-pr 5322 |
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-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 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-iun 4914 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 |
This theorem is referenced by: cnvtrclfv 40062 |
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