Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 5969 | . 2 ⊢ Rel ◡∪ 𝑥 ∈ 𝐴 𝐵 | |
2 | reliun 5691 | . . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel ◡𝐵) | |
3 | relcnv 5969 | . . . 4 ⊢ Rel ◡𝐵 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → Rel ◡𝐵) |
5 | 2, 4 | mprgbir 3155 | . 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
6 | vex 3499 | . . . . . 6 ⊢ 𝑦 ∈ V | |
7 | vex 3499 | . . . . . 6 ⊢ 𝑧 ∈ V | |
8 | 6, 7 | opelcnv 5754 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ◡𝐵 ↔ 〈𝑧, 𝑦〉 ∈ 𝐵) |
9 | 8 | bicomi 226 | . . . 4 ⊢ (〈𝑧, 𝑦〉 ∈ 𝐵 ↔ 〈𝑦, 𝑧〉 ∈ ◡𝐵) |
10 | 9 | rexbii 3249 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 〈𝑧, 𝑦〉 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ ◡𝐵) |
11 | 6, 7 | opelcnv 5754 | . . . 4 ⊢ (〈𝑦, 𝑧〉 ∈ ◡∪ 𝑥 ∈ 𝐴 𝐵 ↔ 〈𝑧, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
12 | eliun 4925 | . . . 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 4925 | . . 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 5665 | 1 ⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∃wrex 3141 〈cop 4575 ∪ ciun 4921 ◡ccnv 5556 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-iun 4923 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: cnvtrclfv 40076 |
Copyright terms: Public domain | W3C validator |