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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 38338 | . 2 ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ∧ Rel 𝑅)) | |
2 | alrmomodm 38341 | . . 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 847 ∀wal 1535 = wceq 1537 ∃*wmo 2536 ∀wral 3059 ∃*wrmo 3377 ∩ cin 3962 ∅c0 4339 class class class wbr 5148 dom cdm 5689 Rel wrel 5694 [cec 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 |
This theorem is referenced by: cosscnvssid5 38460 |
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