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Mirrors > Home > MPE Home > Th. List > funimass2 | Structured version Visualization version GIF version |
Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.) |
Ref | Expression |
---|---|
funimass2 | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimacnv 6438 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹)) | |
2 | 1 | sseq2d 4002 | . . 3 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) ↔ (𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹))) |
3 | inss1 4208 | . . . 4 ⊢ (𝐵 ∩ ran 𝐹) ⊆ 𝐵 | |
4 | sstr2 3977 | . . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)) | |
5 | 3, 4 | mpi 20 | . . 3 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹 “ 𝐴) ⊆ 𝐵) |
6 | 2, 5 | syl6bi 255 | . 2 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵)) |
7 | imass2 5968 | . 2 ⊢ (𝐴 ⊆ (◡𝐹 “ 𝐵) → (𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵))) | |
8 | 6, 7 | impel 508 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∩ cin 3938 ⊆ wss 3939 ◡ccnv 5557 ran crn 5559 “ cima 5561 Fun wfun 6352 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-fun 6360 |
This theorem is referenced by: fvimacnvi 6825 lmhmlsp 19824 2ndcomap 22069 tgqtop 22323 kqreglem1 22352 fmfnfmlem4 22568 fmucnd 22904 cfilucfil 23172 |
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