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Mirrors > Home > MPE Home > Th. List > Mathboxes > brrangeg | Structured version Visualization version GIF version |
Description: Closed form of brrange 33395. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
brrangeg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5068 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎Range𝑏 ↔ 𝐴Range𝑏)) | |
2 | rneq 5805 | . . . 4 ⊢ (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴) | |
3 | 2 | eqeq2d 2832 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = ran 𝑎 ↔ 𝑏 = ran 𝐴)) |
4 | 1, 3 | bibi12d 348 | . 2 ⊢ (𝑎 = 𝐴 → ((𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) ↔ (𝐴Range𝑏 ↔ 𝑏 = ran 𝐴))) |
5 | breq2 5069 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐴Range𝑏 ↔ 𝐴Range𝐵)) | |
6 | eqeq1 2825 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = ran 𝐴 ↔ 𝐵 = ran 𝐴)) | |
7 | 5, 6 | bibi12d 348 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐴Range𝑏 ↔ 𝑏 = ran 𝐴) ↔ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴))) |
8 | vex 3497 | . . 3 ⊢ 𝑎 ∈ V | |
9 | vex 3497 | . . 3 ⊢ 𝑏 ∈ V | |
10 | 8, 9 | brrange 33395 | . 2 ⊢ (𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) |
11 | 4, 7, 10 | vtocl2g 3571 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ran crn 5555 Rangecrange 33305 |
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 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-symdif 4218 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-eprel 5464 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fo 6360 df-fv 6362 df-1st 7688 df-2nd 7689 df-txp 33315 df-image 33325 df-range 33329 |
This theorem is referenced by: (None) |
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