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| Mirrors > Home > MPE Home > Th. List > eliman0 | Structured version Visualization version GIF version | ||
| Description: A nonempty function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.) | 
| Ref | Expression | 
|---|---|
| eliman0 | ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ (𝐹‘𝐴) = ∅) → (𝐹‘𝐴) ∈ (𝐹 “ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvbr0 6935 | . . . . 5 ⊢ (𝐴𝐹(𝐹‘𝐴) ∨ (𝐹‘𝐴) = ∅) | |
| 2 | orcom 871 | . . . . 5 ⊢ ((𝐴𝐹(𝐹‘𝐴) ∨ (𝐹‘𝐴) = ∅) ↔ ((𝐹‘𝐴) = ∅ ∨ 𝐴𝐹(𝐹‘𝐴))) | |
| 3 | 1, 2 | mpbi 230 | . . . 4 ⊢ ((𝐹‘𝐴) = ∅ ∨ 𝐴𝐹(𝐹‘𝐴)) | 
| 4 | 3 | ori 862 | . . 3 ⊢ (¬ (𝐹‘𝐴) = ∅ → 𝐴𝐹(𝐹‘𝐴)) | 
| 5 | breq1 5146 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹(𝐹‘𝐴) ↔ 𝐴𝐹(𝐹‘𝐴))) | |
| 6 | 5 | rspcev 3622 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴𝐹(𝐹‘𝐴)) → ∃𝑥 ∈ 𝐵 𝑥𝐹(𝐹‘𝐴)) | 
| 7 | 4, 6 | sylan2 593 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ (𝐹‘𝐴) = ∅) → ∃𝑥 ∈ 𝐵 𝑥𝐹(𝐹‘𝐴)) | 
| 8 | fvex 6919 | . . 3 ⊢ (𝐹‘𝐴) ∈ V | |
| 9 | 8 | elima 6083 | . 2 ⊢ ((𝐹‘𝐴) ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹(𝐹‘𝐴)) | 
| 10 | 7, 9 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ (𝐹‘𝐴) = ∅) → (𝐹‘𝐴) ∈ (𝐹 “ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∅c0 4333 class class class wbr 5143 “ cima 5688 ‘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-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: ovima0 7612 setrec2fun 49211 | 
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