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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 9509 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
3 | isof1o 7304 | . . . 4 ⊢ (𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) → 𝐹:dom 𝐹–1-1-onto→𝐴) | |
4 | f1of1 6819 | . . . 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 6774 | . . . . 5 ⊢ (𝐹:dom 𝐹–1-1→𝐴 → 𝐹:dom 𝐹⟶𝐴) | |
7 | f1dmex 7925 | . . . . 5 ⊢ ((𝐹:dom 𝐹–1-1→𝐴 ∧ 𝐴 ∈ 𝑉) → dom 𝐹 ∈ V) | |
8 | fex 7212 | . . . . 5 ⊢ ((𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ∈ V) → 𝐹 ∈ V) | |
9 | 6, 7, 8 | syl2an2r 683 | . . . 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 9505 | . . 3 ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅) |
13 | 0ex 5300 | . . 3 ⊢ ∅ ∈ V | |
14 | 12, 13 | eqeltrdi 2840 | . 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 1541 ∈ wcel 2106 Vcvv 3473 ∅c0 4318 E cep 5572 Se wse 5622 We wwe 5623 dom cdm 5669 ⟶wf 6528 –1-1→wf1 6529 –1-1-onto→wf1o 6531 Isom wiso 6533 OrdIsocoi 9486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-oi 9487 |
This theorem is referenced by: oion 9513 oien 9515 cantnfval 9645 wemapwe 9674 finnisoeu 10090 cofsmo 10246 |
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