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Mirrors > Home > MPE Home > Th. List > Mathboxes > ttrclexg | Structured version Visualization version GIF version |
Description: If 𝑅 is a set, then so is t++𝑅. (Contributed by Scott Fenton, 26-Oct-2024.) |
Ref | Expression |
---|---|
ttrclexg | ⊢ (𝑅 ∈ 𝑉 → t++𝑅 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmexg 7700 | . . 3 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
2 | rnexg 7701 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V) | |
3 | 1, 2 | xpexd 7555 | . 2 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 × ran 𝑅) ∈ V) |
4 | relttrcl 33537 | . . . . 5 ⊢ Rel t++𝑅 | |
5 | relssdmrn 6147 | . . . . 5 ⊢ (Rel t++𝑅 → t++𝑅 ⊆ (dom t++𝑅 × ran t++𝑅)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ t++𝑅 ⊆ (dom t++𝑅 × ran t++𝑅) |
7 | dmttrcl 33546 | . . . . 5 ⊢ dom t++𝑅 = dom 𝑅 | |
8 | rnttrcl 33547 | . . . . 5 ⊢ ran t++𝑅 = ran 𝑅 | |
9 | 7, 8 | xpeq12i 5594 | . . . 4 ⊢ (dom t++𝑅 × ran t++𝑅) = (dom 𝑅 × ran 𝑅) |
10 | 6, 9 | sseqtri 3952 | . . 3 ⊢ t++𝑅 ⊆ (dom 𝑅 × ran 𝑅) |
11 | 10 | a1i 11 | . 2 ⊢ (𝑅 ∈ 𝑉 → t++𝑅 ⊆ (dom 𝑅 × ran 𝑅)) |
12 | 3, 11 | ssexd 5232 | 1 ⊢ (𝑅 ∈ 𝑉 → t++𝑅 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Vcvv 3421 ⊆ wss 3881 × cxp 5564 dom cdm 5566 ran crn 5567 Rel wrel 5571 t++cttrcl 33532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-int 4875 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-oadd 8227 df-ttrcl 33533 |
This theorem is referenced by: dfttrcl2 33549 |
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