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Mirrors > Home > MPE Home > Th. List > wemoiso | Structured version Visualization version GIF version |
Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 8819. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
wemoiso | ⊢ (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . . . . . 6 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑅 We 𝐴) | |
2 | vex 3176 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
3 | isof1o 6473 | . . . . . . . . . 10 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴–1-1-onto→𝐵) | |
4 | f1of 6050 | . . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵) | |
5 | 3, 4 | syl 17 | . . . . . . . . 9 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝑓:𝐴⟶𝐵) |
6 | dmfex 7017 | . . . . . . . . 9 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴⟶𝐵) → 𝐴 ∈ V) | |
7 | 2, 5, 6 | sylancr 694 | . . . . . . . 8 ⊢ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐴 ∈ V) |
8 | 7 | ad2antrl 760 | . . . . . . 7 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐴 ∈ V) |
9 | exse 5002 | . . . . . . 7 ⊢ (𝐴 ∈ V → 𝑅 Se 𝐴) | |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑅 Se 𝐴) |
11 | 1, 10 | jca 553 | . . . . 5 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴)) |
12 | weisoeq 6505 | . . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔) | |
13 | 11, 12 | sylancom 698 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝑓 = 𝑔) |
14 | 13 | ex 449 | . . 3 ⊢ (𝑅 We 𝐴 → ((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔)) |
15 | 14 | alrimivv 1843 | . 2 ⊢ (𝑅 We 𝐴 → ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔)) |
16 | isoeq1 6467 | . . 3 ⊢ (𝑓 = 𝑔 → (𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵))) | |
17 | 16 | mo4 2505 | . 2 ⊢ (∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ ∀𝑓∀𝑔((𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑔 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → 𝑓 = 𝑔)) |
18 | 15, 17 | sylibr 223 | 1 ⊢ (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom 𝑅, 𝑆 (𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 ∃*wmo 2459 Vcvv 3173 Se wse 4995 We wwe 4996 ⟶wf 5800 –1-1-onto→wf1o 5803 Isom wiso 5805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 |
This theorem is referenced by: fzisoeu 38455 |
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