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Mirrors > Home > MPE Home > Th. List > Mathboxes > brrangeg | Structured version Visualization version GIF version |
Description: Closed form of brrange 35439. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
brrangeg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5144 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎Range𝑏 ↔ 𝐴Range𝑏)) | |
2 | rneq 5929 | . . . 4 ⊢ (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴) | |
3 | 2 | eqeq2d 2737 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = ran 𝑎 ↔ 𝑏 = ran 𝐴)) |
4 | 1, 3 | bibi12d 345 | . 2 ⊢ (𝑎 = 𝐴 → ((𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) ↔ (𝐴Range𝑏 ↔ 𝑏 = ran 𝐴))) |
5 | breq2 5145 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐴Range𝑏 ↔ 𝐴Range𝐵)) | |
6 | eqeq1 2730 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = ran 𝐴 ↔ 𝐵 = ran 𝐴)) | |
7 | 5, 6 | bibi12d 345 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐴Range𝑏 ↔ 𝑏 = ran 𝐴) ↔ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴))) |
8 | vex 3472 | . . 3 ⊢ 𝑎 ∈ V | |
9 | vex 3472 | . . 3 ⊢ 𝑏 ∈ V | |
10 | 8, 9 | brrange 35439 | . 2 ⊢ (𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) |
11 | 4, 7, 10 | vtocl2g 3557 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ran crn 5670 Rangecrange 35349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-symdif 4237 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-1st 7974 df-2nd 7975 df-txp 35359 df-image 35369 df-range 35373 |
This theorem is referenced by: (None) |
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