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Mirrors > Home > MPE Home > Th. List > Mathboxes > brdomaing | Structured version Visualization version GIF version |
Description: Closed form of brdomain 33389. (Contributed by Scott Fenton, 2-May-2014.) |
Ref | Expression |
---|---|
brdomaing | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5062 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎Domain𝑏 ↔ 𝐴Domain𝑏)) | |
2 | dmeq 5767 | . . . 4 ⊢ (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴) | |
3 | 2 | eqeq2d 2832 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = dom 𝑎 ↔ 𝑏 = dom 𝐴)) |
4 | 1, 3 | bibi12d 348 | . 2 ⊢ (𝑎 = 𝐴 → ((𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴))) |
5 | breq2 5063 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐴Domain𝑏 ↔ 𝐴Domain𝐵)) | |
6 | eqeq1 2825 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = dom 𝐴 ↔ 𝐵 = dom 𝐴)) | |
7 | 5, 6 | bibi12d 348 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴))) |
8 | vex 3498 | . . 3 ⊢ 𝑎 ∈ V | |
9 | vex 3498 | . . 3 ⊢ 𝑏 ∈ V | |
10 | 8, 9 | brdomain 33389 | . 2 ⊢ (𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎) |
11 | 4, 7, 10 | vtocl2g 3572 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 dom cdm 5550 Domaincdomain 33299 |
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-pow 5259 ax-pr 5322 ax-un 7455 |
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-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-symdif 4219 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-eprel 5460 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fo 6356 df-fv 6358 df-1st 7683 df-2nd 7684 df-txp 33310 df-image 33320 df-domain 33323 |
This theorem is referenced by: (None) |
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