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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 8984 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
3 | isof1o 7065 | . . . 4 ⊢ (𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴) | |
4 | f1of1 6607 | . . . 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 6568 | . . . . 5 ⊢ (𝐹:dom 𝐹–1-1→𝐴 → 𝐹:dom 𝐹⟶𝐴) | |
7 | f1dmex 7647 | . . . . 5 ⊢ ((𝐹:dom 𝐹–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → dom 𝐹 ∈ V) | |
8 | fex 6980 | . . . . 5 ⊢ ((𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
9 | 6, 7, 8 | syl2an2r 681 | . . . 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 8980 | . . 3 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅) |
13 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
14 | 12, 13 | syl6eqel 2918 | . 2 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 ∈ V) |
15 | 11, 14 | pm2.61d1 181 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 E cep 5457 Se wse 5505 We wwe 5506 dom cdm 5548 ⟶wf 6344 –1-1→wf1 6345 –1-1-onto→wf1o 6347 Isom wiso 6349 OrdIsocoi 8961 |
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-ext 2790 ax-rep 5181 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-3or 1080 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-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-wrecs 7936 df-recs 7997 df-oi 8962 |
This theorem is referenced by: oion 8988 oien 8990 cantnfval 9119 wemapwe 9148 finnisoeu 9527 cofsmo 9679 |
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