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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 6672 | . . . . 5 ⊢ (𝐴𝐹(𝐹‘𝐴) ∨ (𝐹‘𝐴) = ∅) | |
2 | orcom 867 | . . . . 5 ⊢ ((𝐴𝐹(𝐹‘𝐴) ∨ (𝐹‘𝐴) = ∅) ↔ ((𝐹‘𝐴) = ∅ ∨ 𝐴𝐹(𝐹‘𝐴))) | |
3 | 1, 2 | mpbi 233 | . . . 4 ⊢ ((𝐹‘𝐴) = ∅ ∨ 𝐴𝐹(𝐹‘𝐴)) |
4 | 3 | ori 858 | . . 3 ⊢ (¬ (𝐹‘𝐴) = ∅ → 𝐴𝐹(𝐹‘𝐴)) |
5 | breq1 5033 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹(𝐹‘𝐴) ↔ 𝐴𝐹(𝐹‘𝐴))) | |
6 | 5 | rspcev 3571 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴𝐹(𝐹‘𝐴)) → ∃𝑥 ∈ 𝐵 𝑥𝐹(𝐹‘𝐴)) |
7 | 4, 6 | sylan2 595 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ (𝐹‘𝐴) = ∅) → ∃𝑥 ∈ 𝐵 𝑥𝐹(𝐹‘𝐴)) |
8 | fvex 6658 | . . 3 ⊢ (𝐹‘𝐴) ∈ V | |
9 | 8 | elima 5901 | . 2 ⊢ ((𝐹‘𝐴) ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹(𝐹‘𝐴)) |
10 | 7, 9 | sylibr 237 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ (𝐹‘𝐴) = ∅) → (𝐹‘𝐴) ∈ (𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ∅c0 4243 class class class wbr 5030 “ cima 5522 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fv 6332 |
This theorem is referenced by: ovima0 7307 setrec2fun 45222 |
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