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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 7369 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) | |
2 | 1 | adantr 474 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ( I ↾ 𝐴) ∈ V) |
3 | xrnresex 34707 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ ( I ↾ 𝐴) ∈ V) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V) | |
4 | 2, 3 | mpd3an3 1590 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 Vcvv 3414 I cid 5251 ↾ cres 5348 ⋉ cxrn 34518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fo 6133 df-fv 6135 df-1st 7433 df-2nd 7434 df-xrn 34676 |
This theorem is referenced by: (None) |
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