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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elrnressn | Structured version Visualization version GIF version |
Description: Element of the range of a restriction to a singleton. (Contributed by Peter Mazsa, 12-Jun-2024.) |
Ref | Expression |
---|---|
elrnressn | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrnres 38252 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ ∃𝑥 ∈ {𝐴}𝑥𝑅𝐵)) | |
2 | breq1 5150 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐵 ↔ 𝐴𝑅𝐵)) | |
3 | 2 | rexsng 4680 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑥𝑅𝐵 ↔ 𝐴𝑅𝐵)) |
4 | 1, 3 | sylan9bbr 510 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2105 ∃wrex 3067 {csn 4630 class class class wbr 5147 ran crn 5689 ↾ cres 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 |
This theorem is referenced by: refressn 38424 |
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