![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > brovmptimex | Structured version Visualization version GIF version |
Description: If a binary relation holds and the relation is the value of a binary operation built with maps-to, then the arguments to that operation are sets. (Contributed by RP, 22-May-2021.) |
Ref | Expression |
---|---|
brovmptimex.mpt | ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻) |
brovmptimex.br | ⊢ (𝜑 → 𝐴𝑅𝐵) |
brovmptimex.ov | ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷)) |
Ref | Expression |
---|---|
brovmptimex | ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brovmptimex.ov | . . 3 ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷)) | |
2 | brovmptimex.br | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
3 | 1, 2 | breqdi 5045 | . 2 ⊢ (𝜑 → 𝐴(𝐶𝐹𝐷)𝐵) |
4 | brne0 5080 | . 2 ⊢ (𝐴(𝐶𝐹𝐷)𝐵 → (𝐶𝐹𝐷) ≠ ∅) | |
5 | brovmptimex.mpt | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻) | |
6 | 5 | reldmmpo 7264 | . . . 4 ⊢ Rel dom 𝐹 |
7 | 6 | ovprc 7173 | . . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶𝐹𝐷) = ∅) |
8 | 7 | necon1ai 3014 | . 2 ⊢ ((𝐶𝐹𝐷) ≠ ∅ → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
9 | 3, 4, 8 | 3syl 18 | 1 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 class class class wbr 5030 (class class class)co 7135 ∈ cmpo 7137 |
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-pow 5231 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-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-rel 5526 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: brovmptimex1 40731 brovmptimex2 40732 |
Copyright terms: Public domain | W3C validator |