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Mirrors > Home > MPE Home > Th. List > Mathboxes > alrmomodm | Structured version Visualization version GIF version |
Description: Equivalence of an "at most one" and an "at most one" restricted to the domain inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.) |
Ref | Expression |
---|---|
alrmomodm | ⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 3371 | . . 3 ⊢ (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥)) | |
2 | brres 5986 | . . . . . 6 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥))) | |
3 | 2 | elv 3475 | . . . . 5 ⊢ (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥)) |
4 | resdm 6024 | . . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅) | |
5 | 4 | breqd 5153 | . . . . 5 ⊢ (Rel 𝑅 → (𝑢(𝑅 ↾ dom 𝑅)𝑥 ↔ 𝑢𝑅𝑥)) |
6 | 3, 5 | bitr3id 285 | . . . 4 ⊢ (Rel 𝑅 → ((𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥) ↔ 𝑢𝑅𝑥)) |
7 | 6 | mobidv 2538 | . . 3 ⊢ (Rel 𝑅 → (∃*𝑢(𝑢 ∈ dom 𝑅 ∧ 𝑢𝑅𝑥) ↔ ∃*𝑢 𝑢𝑅𝑥)) |
8 | 1, 7 | bitrid 283 | . 2 ⊢ (Rel 𝑅 → (∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∃*𝑢 𝑢𝑅𝑥)) |
9 | 8 | albidv 1916 | 1 ⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 ∈ wcel 2099 ∃*wmo 2527 ∃*wrmo 3370 Vcvv 3469 class class class wbr 5142 dom cdm 5672 ↾ cres 5674 Rel wrel 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2529 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rmo 3371 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-dm 5682 df-res 5684 |
This theorem is referenced by: inecmo3 37756 |
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