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Mirrors > Home > MPE Home > Th. List > brovmpt2ex | Structured version Visualization version GIF version |
Description: A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.) |
Ref | Expression |
---|---|
brovmpt2ex.1 | ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) |
Ref | Expression |
---|---|
brovmpt2ex | ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brovmpt2ex.1 | . 2 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) | |
2 | 1 | relmpt2opab 7259 | . . 3 ⊢ Rel (𝑉𝑂𝐸) |
3 | 2 | a1i 11 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Rel (𝑉𝑂𝐸)) |
4 | 1, 3 | brovex 7348 | 1 ⊢ (𝐹(𝑉𝑂𝐸)𝑃 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 {copab 4712 Rel wrel 5119 (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 ax-un 6949 |
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-sbc 3436 df-csb 3534 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-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: (None) |
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