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| Mirrors > Home > MPE Home > Th. List > funfvima2d | Structured version Visualization version GIF version | ||
| Description: A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.) | 
| Ref | Expression | 
|---|---|
| funfvima2d.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | 
| Ref | Expression | 
|---|---|
| funfvima2d | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | funfvima2d.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 2 | 1 | ffund 6740 | . . 3 ⊢ (𝜑 → Fun 𝐹) | 
| 3 | ssidd 4007 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) | |
| 4 | 1 | fdmd 6746 | . . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴) | 
| 5 | 3, 4 | sseqtrrd 4021 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) | 
| 6 | funfvima2 7251 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))) | |
| 7 | 2, 5, 6 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))) | 
| 8 | 7 | imp 406 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 dom cdm 5685 “ cima 5688 Fun wfun 6555 ⟶wf 6557 ‘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-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-f 6565 df-fv 6569 | 
| This theorem is referenced by: imo72b2lem1 44182 fundcmpsurbijinjpreimafv 47394 | 
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