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| Mirrors > Home > MPE Home > Th. List > funimass3 | Structured version Visualization version GIF version | ||
| Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 7073 would be the special case of 𝐴 being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.) |
| Ref | Expression |
|---|---|
| funimass3 | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funimass4 6973 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
| 2 | ssel 3977 | . . . . . 6 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹)) | |
| 3 | fvimacnv 7073 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵))) | |
| 4 | 3 | ex 412 | . . . . . 6 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵)))) |
| 5 | 2, 4 | syl9r 78 | . . . . 5 ⊢ (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵))))) |
| 6 | 5 | imp31 417 | . . . 4 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵))) |
| 7 | 6 | ralbidva 3176 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵))) |
| 8 | 1, 7 | bitrd 279 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵))) |
| 9 | dfss3 3972 | . 2 ⊢ (𝐴 ⊆ (◡𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵)) | |
| 10 | 8, 9 | bitr4di 289 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ◡ccnv 5684 dom cdm 5685 “ cima 5688 Fun wfun 6555 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: funimass5 7075 funconstss 7076 fvimacnvALT 7077 r0weon 10052 iscnp3 23252 cnpnei 23272 cnclsi 23280 cncls 23282 cncnp 23288 1stccnp 23470 txcnpi 23616 xkoco2cn 23666 xkococnlem 23667 basqtop 23719 kqnrmlem1 23751 kqnrmlem2 23752 reghmph 23801 nrmhmph 23802 elfm3 23958 rnelfm 23961 symgtgp 24114 tgpconncompeqg 24120 eltsms 24141 ucnprima 24291 plyco0 26231 plyeq0 26250 xrlimcnp 27011 rinvf1o 32640 xppreima 32655 rhmpreimacnlem 33883 cvmliftmolem1 35286 cvmlift2lem9 35316 cvmlift3lem6 35329 mclsppslem 35588 |
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