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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 41266 | . 2 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐷 ∈ V)) |
5 | 4 | simprd 499 | 1 ⊢ (𝜑 → 𝐷 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Vcvv 3401 class class class wbr 5043 (class class class)co 7202 ∈ cmpo 7204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-dm 5550 df-iota 6327 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 |
This theorem is referenced by: ntrneibex 41312 |
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