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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 37045 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ ∃𝑥 ∈ {𝐴}𝑥𝑅𝐵)) | |
2 | breq1 5147 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐵 ↔ 𝐴𝑅𝐵)) | |
3 | 2 | rexsng 4674 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑥𝑅𝐵 ↔ 𝐴𝑅𝐵)) |
4 | 1, 3 | sylan9bbr 512 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∃wrex 3071 {csn 4624 class class class wbr 5144 ran crn 5673 ↾ cres 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5678 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 |
This theorem is referenced by: refressn 37219 |
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