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Mirrors > Home > MPE Home > Th. List > Mathboxes > brdomaing | Structured version Visualization version GIF version |
Description: Closed form of brdomain 33881. (Contributed by Scott Fenton, 2-May-2014.) |
Ref | Expression |
---|---|
brdomaing | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5034 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎Domain𝑏 ↔ 𝐴Domain𝑏)) | |
2 | dmeq 5747 | . . . 4 ⊢ (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴) | |
3 | 2 | eqeq2d 2750 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = dom 𝑎 ↔ 𝑏 = dom 𝐴)) |
4 | 1, 3 | bibi12d 349 | . 2 ⊢ (𝑎 = 𝐴 → ((𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴))) |
5 | breq2 5035 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐴Domain𝑏 ↔ 𝐴Domain𝐵)) | |
6 | eqeq1 2743 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = dom 𝐴 ↔ 𝐵 = dom 𝐴)) | |
7 | 5, 6 | bibi12d 349 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴))) |
8 | vex 3403 | . . 3 ⊢ 𝑎 ∈ V | |
9 | vex 3403 | . . 3 ⊢ 𝑏 ∈ V | |
10 | 8, 9 | brdomain 33881 | . 2 ⊢ (𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎) |
11 | 4, 7, 10 | vtocl2g 3476 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5031 dom cdm 5526 Domaincdomain 33791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-symdif 4134 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-eprel 5435 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-fo 6346 df-fv 6348 df-1st 7717 df-2nd 7718 df-txp 33802 df-image 33812 df-domain 33815 |
This theorem is referenced by: (None) |
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