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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 6124 | . 2 ⊢ Rel ◡∪ 𝑥 ∈ 𝐴 𝐵 | |
2 | reliun 5828 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel ◡𝐵) | |
3 | relcnv 6124 | . . . 4 ⊢ Rel ◡𝐵 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → Rel ◡𝐵) |
5 | 2, 4 | mprgbir 3065 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
6 | vex 3481 | . . . . . 6 ⊢ 𝑦 ∈ V | |
7 | vex 3481 | . . . . . 6 ⊢ 𝑧 ∈ V | |
8 | 6, 7 | opelcnv 5894 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ◡𝐵 ↔ 〈𝑧, 𝑦〉 ∈ 𝐵) |
9 | 8 | bicomi 224 | . . . 4 ⊢ (〈𝑧, 𝑦〉 ∈ 𝐵 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐵) |
10 | 9 | rexbii 3091 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ ◡𝐵) |
11 | 6, 7 | opelcnv 5894 | . . . 4 ⊢ (〈𝑦, 𝑧〉 ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
12 | eliun 4999 | . . . 4 ⊢ (〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵) | |
13 | 11, 12 | bitri 275 | . . 3 ⊢ (〈𝑦, 𝑧〉 ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵) |
14 | eliun 4999 | . . 3 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ ◡𝐵) | |
15 | 10, 13, 14 | 3bitr4i 303 | . 2 ⊢ (〈𝑦, 𝑧〉 ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ 〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝐵) |
16 | 1, 5, 15 | eqrelriiv 5802 | 1 ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 ∃wrex 3067 〈cop 4636 ∪ ciun 4995 ◡ccnv 5687 Rel wrel 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-iun 4997 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 |
This theorem is referenced by: cnvtrclfv 43713 |
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