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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 7907. (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 7276 | . . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐹 Isom 𝑆, 𝑅 (𝐵, 𝐴)) | |
3 | isotr 7282 | . . . 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 598 | . . 3 ⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)) → (◡𝐹 ∘ 𝐺) Isom 𝑅, 𝑅 (𝐴, 𝐴)) |
5 | weniso 7300 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ (◡𝐹 ∘ 𝐺) Isom 𝑅, 𝑅 (𝐴, 𝐴)) → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) | |
6 | 5 | 3expa 1119 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (◡𝐹 ∘ 𝐺) Isom 𝑅, 𝑅 (𝐴, 𝐴)) → (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴)) |
7 | 4, 6 | sylan2 594 | . 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 770 | . . . 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 7269 | . . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵) | |
10 | f1of1 6784 | . . . 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 772 | . . . 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 7269 | . . . 4 ⊢ (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐺:𝐴–1-1-onto→𝐵) | |
14 | f1of1 6784 | . . . 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 7248 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) | |
17 | 11, 15, 16 | syl2anc 585 | . 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 257 | 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 205 ∧ wa 397 = wceq 1542 I cid 5531 Se wse 5587 We wwe 5588 ◡ccnv 5633 ↾ cres 5636 ∘ ccom 5638 –1-1→wf1 6494 –1-1-onto→wf1o 6496 Isom wiso 6498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 |
This theorem is referenced by: weisoeq2 7302 wemoiso 7907 oieu 9480 |
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