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Mirrors > Home > MPE Home > Th. List > Mathboxes > brdomaing | Structured version Visualization version GIF version |
Description: Closed form of brdomain 34564. (Contributed by Scott Fenton, 2-May-2014.) |
Ref | Expression |
---|---|
brdomaing | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5109 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎Domain𝑏 ↔ 𝐴Domain𝑏)) | |
2 | dmeq 5860 | . . . 4 ⊢ (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴) | |
3 | 2 | eqeq2d 2744 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = dom 𝑎 ↔ 𝑏 = dom 𝐴)) |
4 | 1, 3 | bibi12d 346 | . 2 ⊢ (𝑎 = 𝐴 → ((𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴))) |
5 | breq2 5110 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐴Domain𝑏 ↔ 𝐴Domain𝐵)) | |
6 | eqeq1 2737 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = dom 𝐴 ↔ 𝐵 = dom 𝐴)) | |
7 | 5, 6 | bibi12d 346 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴))) |
8 | vex 3448 | . . 3 ⊢ 𝑎 ∈ V | |
9 | vex 3448 | . . 3 ⊢ 𝑏 ∈ V | |
10 | 8, 9 | brdomain 34564 | . 2 ⊢ (𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎) |
11 | 4, 7, 10 | vtocl2g 3530 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 dom cdm 5634 Domaincdomain 34474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-symdif 4203 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-eprel 5538 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 df-fv 6505 df-1st 7922 df-2nd 7923 df-txp 34485 df-image 34495 df-domain 34498 |
This theorem is referenced by: (None) |
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