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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 7055 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 6957 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | ssel 3976 | . . . . . 6 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹)) | |
3 | fvimacnv 7055 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵))) | |
4 | 3 | ex 414 | . . . . . 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 419 | . . . 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 3971 | . 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 205 ∧ wa 397 ∈ wcel 2107 ∀wral 3062 ⊆ wss 3949 ◡ccnv 5676 dom cdm 5677 “ cima 5680 Fun wfun 6538 ‘cfv 6544 |
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 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 |
This theorem is referenced by: funimass5 7057 funconstss 7058 fvimacnvALT 7059 fimacnvOLD 7073 r0weon 10007 iscnp3 22748 cnpnei 22768 cnclsi 22776 cncls 22778 cncnp 22784 1stccnp 22966 txcnpi 23112 xkoco2cn 23162 xkococnlem 23163 basqtop 23215 kqnrmlem1 23247 kqnrmlem2 23248 reghmph 23297 nrmhmph 23298 elfm3 23454 rnelfm 23457 symgtgp 23610 tgpconncompeqg 23616 eltsms 23637 ucnprima 23787 plyco0 25706 plyeq0 25725 xrlimcnp 26473 rinvf1o 31854 xppreima 31871 rhmpreimacnlem 32864 cvmliftmolem1 34272 cvmlift2lem9 34302 cvmlift3lem6 34315 mclsppslem 34574 |
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