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Mirrors > Home > MPE Home > Th. List > Mathboxes > brrangeg | Structured version Visualization version GIF version |
Description: Closed form of brrange 35401. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
brrangeg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5141 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎Range𝑏 ↔ 𝐴Range𝑏)) | |
2 | rneq 5925 | . . . 4 ⊢ (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴) | |
3 | 2 | eqeq2d 2735 | . . 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 5142 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐴Range𝑏 ↔ 𝐴Range𝐵)) | |
6 | eqeq1 2728 | . . 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 3470 | . . 3 ⊢ 𝑎 ∈ V | |
9 | vex 3470 | . . 3 ⊢ 𝑏 ∈ V | |
10 | 8, 9 | brrange 35401 | . 2 ⊢ (𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) |
11 | 4, 7, 10 | vtocl2g 3555 | 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 5138 ran crn 5667 Rangecrange 35311 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-symdif 4234 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-eprel 5570 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fo 6539 df-fv 6541 df-1st 7968 df-2nd 7969 df-txp 35321 df-image 35331 df-range 35335 |
This theorem is referenced by: (None) |
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