Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovprc | Structured version Visualization version GIF version |
Description: The value of an operation when the one of the arguments is a proper class, analogous to ovprc 7194. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovprc.1 | ⊢ Rel dom 𝐹 |
Ref | Expression |
---|---|
aovprc | ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aov 43340 | . 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | df-br 5067 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
3 | aovprc.1 | . . . . . 6 ⊢ Rel dom 𝐹 | |
4 | 3 | brrelex12i 5607 | . . . . 5 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | 2, 4 | sylbir 237 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
6 | 5 | con3i 157 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
7 | ndmafv 43359 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐹'''〈𝐴, 𝐵〉) = V) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹'''〈𝐴, 𝐵〉) = V) |
9 | 1, 8 | syl5eq 2868 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐹𝐵)) = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 class class class wbr 5066 dom cdm 5555 Rel wrel 5560 '''cafv 43336 ((caov 43337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-aiota 43305 df-dfat 43338 df-afv 43339 df-aov 43340 |
This theorem is referenced by: (None) |
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