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Mirrors > Home > MPE Home > Th. List > elovmpt3imp | Structured version Visualization version GIF version |
Description: If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands must be sets. Remark: a function which is the result of an operation can be regared as operation with 3 operands - therefore the abbreviation "mpt3" is used in the label. (Contributed by AV, 16-May-2019.) |
Ref | Expression |
---|---|
elovmpt3imp.o | ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ 𝐵)) |
Ref | Expression |
---|---|
elovmpt3imp | ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 3921 | . 2 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋𝑂𝑌)‘𝑍) ≠ ∅) | |
2 | ax-1 6 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V))) | |
3 | elovmpt3imp.o | . . . . 5 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ 𝐵)) | |
4 | 3 | mpt2ndm0 6875 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅) |
5 | fveq1 6190 | . . . . 5 ⊢ ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = (∅‘𝑍)) | |
6 | 0fv 6227 | . . . . 5 ⊢ (∅‘𝑍) = ∅ | |
7 | 5, 6 | syl6eq 2672 | . . . 4 ⊢ ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = ∅) |
8 | eqneqall 2805 | . . . 4 ⊢ (((𝑋𝑂𝑌)‘𝑍) = ∅ → (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V))) | |
9 | 4, 7, 8 | 3syl 18 | . . 3 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V))) |
10 | 2, 9 | pm2.61i 176 | . 2 ⊢ (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
11 | 1, 10 | syl 17 | 1 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: elovmpt3rab1 6893 elovmptnn0wrd 13348 |
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