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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrnidresex | Structured version Visualization version GIF version |
Description: Sufficient condition for a range Cartesian product with restricted identity to be a set. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
xrnidresex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resiexg 7608 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ( I ↾ 𝐴) ∈ V) |
3 | xrnresex 35534 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( I ↾ 𝐴) ∈ V) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V) | |
4 | 2, 3 | mpd3an3 1453 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 Vcvv 3492 I cid 5452 ↾ cres 5550 ⋉ cxrn 35333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7678 df-2nd 7679 df-xrn 35503 |
This theorem is referenced by: (None) |
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