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Mirrors > Home > MPE Home > Th. List > oiexg | Structured version Visualization version GIF version |
Description: The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
oicl.1 | ⊢ 𝐹 = OrdIso(𝑅, 𝐴) |
Ref | Expression |
---|---|
oiexg | ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oicl.1 | . . . . 5 ⊢ 𝐹 = OrdIso(𝑅, 𝐴) | |
2 | 1 | ordtype 9291 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
3 | isof1o 7194 | . . . 4 ⊢ (𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴) | |
4 | f1of1 6715 | . . . 4 ⊢ (𝐹:dom 𝐹–1-1-onto→𝐴 → 𝐹:dom 𝐹–1-1→𝐴) | |
5 | 2, 3, 4 | 3syl 18 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹:dom 𝐹–1-1→𝐴) |
6 | f1f 6670 | . . . . 5 ⊢ (𝐹:dom 𝐹–1-1→𝐴 → 𝐹:dom 𝐹⟶𝐴) | |
7 | f1dmex 7799 | . . . . 5 ⊢ ((𝐹:dom 𝐹–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → dom 𝐹 ∈ V) | |
8 | fex 7102 | . . . . 5 ⊢ ((𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
9 | 6, 7, 8 | syl2an2r 682 | . . . 4 ⊢ ((𝐹:dom 𝐹–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) |
10 | 9 | expcom 414 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐹:dom 𝐹–1-1→𝐴 → 𝐹 ∈ V)) |
11 | 5, 10 | syl5 34 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 ∈ V)) |
12 | 1 | oi0 9287 | . . 3 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅) |
13 | 0ex 5231 | . . 3 ⊢ ∅ ∈ V | |
14 | 12, 13 | eqeltrdi 2847 | . 2 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 ∈ V) |
15 | 11, 14 | pm2.61d1 180 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∅c0 4256 E cep 5494 Se wse 5542 We wwe 5543 dom cdm 5589 ⟶wf 6429 –1-1→wf1 6430 –1-1-onto→wf1o 6432 Isom wiso 6434 OrdIsocoi 9268 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-oi 9269 |
This theorem is referenced by: oion 9295 oien 9297 cantnfval 9426 wemapwe 9455 finnisoeu 9869 cofsmo 10025 |
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