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Mirrors > Home > MPE Home > Th. List > Mathboxes > brovmptimex2 | 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 |
---|---|
brovmptimex2 | ⊢ (𝜑 → 𝐷 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brovmptimex.mpt | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐺 ↦ 𝐻) | |
2 | brovmptimex.br | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
3 | brovmptimex.ov | . . 3 ⊢ (𝜑 → 𝑅 = (𝐶𝐹𝐷)) | |
4 | 1, 2, 3 | brovmptimex 40375 | . 2 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
5 | 4 | simprd 498 | 1 ⊢ (𝜑 → 𝐷 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5065 (class class class)co 7155 ∈ cmpo 7157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-dm 5564 df-iota 6313 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 |
This theorem is referenced by: ntrneibex 40421 |
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