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Mirrors > Home > MPE Home > Th. List > wemoiso2 | Structured version Visualization version GIF version |
Description: Thus, there is at most one isomorphism between any two well-ordered sets. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
wemoiso2 | ⊢ (𝑆 We 𝐵 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . . . . 6 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑆 We 𝐵) | |
2 | isof1o 7070 | . . . . . . . . . 10 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴–1-1-onto→𝐵) | |
3 | f1ofo 6616 | . . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵) | |
4 | forn 6587 | . . . . . . . . . 10 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵) | |
5 | 2, 3, 4 | 3syl 18 | . . . . . . . . 9 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ran 𝑓 = 𝐵) |
6 | vex 3497 | . . . . . . . . . 10 ⊢ 𝑓 ∈ V | |
7 | 6 | rnex 7611 | . . . . . . . . 9 ⊢ ran 𝑓 ∈ V |
8 | 5, 7 | eqeltrrdi 2922 | . . . . . . . 8 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐵 ∈ V) |
9 | 8 | ad2antrl 726 | . . . . . . 7 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐵 ∈ V) |
10 | exse 5513 | . . . . . . 7 ⊢ (𝐵 ∈ V → 𝑆 Se 𝐵) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑆 Se 𝐵) |
12 | 1, 11 | jca 514 | . . . . 5 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝑆 We 𝐵 ∧ 𝑆 Se 𝐵)) |
13 | weisoeq2 7103 | . . . . 5 ⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔) | |
14 | 12, 13 | sylancom 590 | . . . 4 ⊢ ((𝑆 We 𝐵 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔) |
15 | 14 | ex 415 | . . 3 ⊢ (𝑆 We 𝐵 → ((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔)) |
16 | 15 | alrimivv 1925 | . 2 ⊢ (𝑆 We 𝐵 → ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔)) |
17 | isoeq1 7064 | . . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) | |
18 | 17 | mo4 2646 | . 2 ⊢ (∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔)) |
19 | 16, 18 | sylibr 236 | 1 ⊢ (𝑆 We 𝐵 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∃*wmo 2616 Vcvv 3494 Se wse 5506 We wwe 5507 ran crn 5550 –onto→wfo 6347 –1-1-onto→wf1o 6348 Isom wiso 6350 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 |
This theorem is referenced by: finnisoeu 9533 |
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