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Mirrors > Home > MPE Home > Th. List > 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 7924 | . . 3 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V) | |
2 | rnexg 7925 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ran 𝑅 ∈ V) | |
3 | 1, 2 | xpexd 7770 | . 2 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 × ran 𝑅) ∈ V) |
4 | relttrcl 9750 | . . . . 5 ⊢ Rel t++𝑅 | |
5 | relssdmrn 6290 | . . . . 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 9759 | . . . . 5 ⊢ dom t++𝑅 = dom 𝑅 | |
8 | rnttrcl 9760 | . . . . 5 ⊢ ran t++𝑅 = ran 𝑅 | |
9 | 7, 8 | xpeq12i 5717 | . . . 4 ⊢ (dom t++𝑅 × ran t++𝑅) = (dom 𝑅 × ran 𝑅) |
10 | 6, 9 | sseqtri 4032 | . . 3 ⊢ t++𝑅 ⊆ (dom 𝑅 × ran 𝑅) |
11 | 10 | a1i 11 | . 2 ⊢ (𝑅 ∈ 𝑉 → t++𝑅 ⊆ (dom 𝑅 × ran 𝑅)) |
12 | 3, 11 | ssexd 5330 | 1 ⊢ (𝑅 ∈ 𝑉 → t++𝑅 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 × cxp 5687 dom cdm 5689 ran crn 5690 Rel wrel 5694 t++cttrcl 9745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-ttrcl 9746 |
This theorem is referenced by: dfttrcl2 9762 |
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