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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 6122 | . 2 ⊢ Rel ◡∪ 𝑥 ∈ 𝐴 𝐵 | |
| 2 | reliun 5826 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel ◡𝐵) | |
| 3 | relcnv 6122 | . . . 4 ⊢ Rel ◡𝐵 | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → Rel ◡𝐵) |
| 5 | 2, 4 | mprgbir 3068 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
| 6 | vex 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 7 | vex 3484 | . . . . . 6 ⊢ 𝑧 ∈ V | |
| 8 | 6, 7 | opelcnv 5892 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ◡𝐵 ↔ 〈𝑧, 𝑦〉 ∈ 𝐵) |
| 9 | 8 | bicomi 224 | . . . 4 ⊢ (〈𝑧, 𝑦〉 ∈ 𝐵 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐵) |
| 10 | 9 | rexbii 3094 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ ◡𝐵) |
| 11 | 6, 7 | opelcnv 5892 | . . . 4 ⊢ (〈𝑦, 𝑧〉 ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
| 12 | eliun 4995 | . . . 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 4995 | . . 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 5800 | 1 ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∃wrex 3070 〈cop 4632 ∪ ciun 4991 ◡ccnv 5684 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: cnvtrclfv 43737 |
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