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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inecmo3 | Structured version Visualization version GIF version |
Description: Equivalence of a double universal quantification restricted to the domain and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.) |
Ref | Expression |
---|---|
inecmo3 | ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inecmo2 38312 | . 2 ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ∧ Rel 𝑅)) | |
2 | alrmomodm 38315 | . . 3 ⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)) | |
3 | 2 | pm5.32ri 575 | . 2 ⊢ ((∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅)) |
4 | 1, 3 | bitri 275 | 1 ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 846 ∀wal 1535 = wceq 1537 ∃*wmo 2541 ∀wral 3067 ∃*wrmo 3387 ∩ cin 3975 ∅c0 4352 class class class wbr 5166 dom cdm 5700 Rel wrel 5705 [cec 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rmo 3388 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-rel 5707 df-cnv 5708 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ec 8765 |
This theorem is referenced by: cosscnvssid5 38434 |
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