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| Mirrors > Home > MPE Home > Th. List > isofr | Structured version Visualization version GIF version | ||
| Description: An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| isofr | ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isocnv 7351 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴)) | |
| 2 | id 22 | . . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴)) | |
| 3 | isof1o 7344 | . . . . 5 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻:𝐵–1-1-onto→𝐴) | |
| 4 | f1ofun 6849 | . . . . 5 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → Fun ◡𝐻) | |
| 5 | vex 3483 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 6 | 5 | funimaex 6654 | . . . . 5 ⊢ (Fun ◡𝐻 → (◡𝐻 “ 𝑥) ∈ V) | 
| 7 | 3, 4, 6 | 3syl 18 | . . . 4 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → (◡𝐻 “ 𝑥) ∈ V) | 
| 8 | 2, 7 | isofrlem 7361 | . . 3 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → (𝑅 Fr 𝐴 → 𝑆 Fr 𝐵)) | 
| 9 | 1, 8 | syl 17 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 → 𝑆 Fr 𝐵)) | 
| 10 | id 22 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | |
| 11 | isof1o 7344 | . . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵) | |
| 12 | f1ofun 6849 | . . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻) | |
| 13 | 5 | funimaex 6654 | . . . 4 ⊢ (Fun 𝐻 → (𝐻 “ 𝑥) ∈ V) | 
| 14 | 11, 12, 13 | 3syl 18 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 “ 𝑥) ∈ V) | 
| 15 | 10, 14 | isofrlem 7361 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴)) | 
| 16 | 9, 15 | impbid 212 | 1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 Vcvv 3479 Fr wfr 5633 ◡ccnv 5683 “ cima 5687 Fun wfun 6554 –1-1-onto→wf1o 6559 Isom wiso 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-fr 5636 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 | 
| This theorem is referenced by: isowe 7370 wofib 9586 isfin1-4 10428 | 
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