Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > weisoeq | Structured version Visualization version GIF version |
Description: Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 7663. (Contributed by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
weisoeq | ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | |
2 | isocnv 7072 | . . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴)) | |
3 | isotr 7078 | . . . 4 ⊢ ((𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴)) → (◡𝐹 ∘ 𝐺) Isom 𝑅, 𝑅 (𝐴, 𝐴)) | |
4 | 1, 2, 3 | syl2anr 596 | . . 3 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → (◡𝐹 ∘ 𝐺) Isom 𝑅, 𝑅 (𝐴, 𝐴)) |
5 | weniso 7096 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ (◡𝐹 ∘ 𝐺) Isom 𝑅, 𝑅 (𝐴, 𝐴)) → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) | |
6 | 5 | 3expa 1110 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (◡𝐹 ∘ 𝐺) Isom 𝑅, 𝑅 (𝐴, 𝐴)) → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) |
7 | 4, 6 | sylan2 592 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) |
8 | simprl 767 | . . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | |
9 | isof1o 7065 | . . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵) | |
10 | f1of1 6607 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) | |
11 | 8, 9, 10 | 3syl 18 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹:𝐴–1-1→𝐵) |
12 | simprr 769 | . . . 4 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | |
13 | isof1o 7065 | . . . 4 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺:𝐴–1-1-onto→𝐵) | |
14 | f1of1 6607 | . . . 4 ⊢ (𝐺:𝐴–1-1-onto→𝐵 → 𝐺:𝐴–1-1→𝐵) | |
15 | 12, 13, 14 | 3syl 18 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐺:𝐴–1-1→𝐵) |
16 | f1eqcocnv 7048 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) | |
17 | 11, 15, 16 | syl2anc 584 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) |
18 | 7, 17 | mpbird 258 | 1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 I cid 5452 Se wse 5505 We wwe 5506 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 –1-1→wf1 6345 –1-1-onto→wf1o 6347 Isom wiso 6349 |
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-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
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-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 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-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 |
This theorem is referenced by: weisoeq2 7098 wemoiso 7663 oieu 8991 |
Copyright terms: Public domain | W3C validator |