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Mirrors > Home > MPE Home > Th. List > Mathboxes > brrangeg | Structured version Visualization version GIF version |
Description: Closed form of brrange 31211. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
brrangeg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4586 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎Range𝑏 ↔ 𝐴Range𝑏)) | |
2 | rneq 5272 | . . . 4 ⊢ (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴) | |
3 | 2 | eqeq2d 2620 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = ran 𝑎 ↔ 𝑏 = ran 𝐴)) |
4 | 1, 3 | bibi12d 334 | . 2 ⊢ (𝑎 = 𝐴 → ((𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) ↔ (𝐴Range𝑏 ↔ 𝑏 = ran 𝐴))) |
5 | breq2 4587 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐴Range𝑏 ↔ 𝐴Range𝐵)) | |
6 | eqeq1 2614 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = ran 𝐴 ↔ 𝐵 = ran 𝐴)) | |
7 | 5, 6 | bibi12d 334 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐴Range𝑏 ↔ 𝑏 = ran 𝐴) ↔ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴))) |
8 | vex 3176 | . . 3 ⊢ 𝑎 ∈ V | |
9 | vex 3176 | . . 3 ⊢ 𝑏 ∈ V | |
10 | 8, 9 | brrange 31211 | . 2 ⊢ (𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) |
11 | 4, 7, 10 | vtocl2g 3243 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ran crn 5039 Rangecrange 31120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-image 31140 df-range 31144 |
This theorem is referenced by: (None) |
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