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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 7117 | . . 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 7110 | . . . . 5 ⊢ (◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴) → ◡𝐻:𝐵–1-1-onto→𝐴) | |
4 | f1ofun 6641 | . . . . 5 ⊢ (◡𝐻:𝐵–1-1-onto→𝐴 → Fun ◡𝐻) | |
5 | vex 3402 | . . . . . 6 ⊢ 𝑥 ∈ V | |
6 | 5 | funimaex 6445 | . . . . 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 7127 | . . 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 7110 | . . . 4 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵) | |
12 | f1ofun 6641 | . . . 4 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → Fun 𝐻) | |
13 | 5 | funimaex 6445 | . . . 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 7127 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴)) |
16 | 9, 15 | impbid 215 | 1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2112 Vcvv 3398 Fr wfr 5491 ◡ccnv 5535 “ cima 5539 Fun wfun 6352 –1-1-onto→wf1o 6357 Isom wiso 6359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-fr 5494 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 |
This theorem is referenced by: isowe 7136 wofib 9139 isfin1-4 9966 |
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