![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ineccnvmo2 | Structured version Visualization version GIF version |
Description: Equivalence of a double universal quantification restricted to the range and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 4-Sep-2021.) |
Ref | Expression |
---|---|
ineccnvmo2 | ⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineccnvmo 37682 | . 2 ⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥) | |
2 | alrmomorn 37683 | . 2 ⊢ (∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥 ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥) | |
3 | 1, 2 | bitri 275 | 1 ⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 844 ∀wal 1531 = wceq 1533 ∃*wmo 2524 ∀wral 3053 ∃*wrmo 3367 ∩ cin 3939 ∅c0 4314 class class class wbr 5138 ◡ccnv 5665 ran crn 5667 [cec 8696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rmo 3368 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-xp 5672 df-rel 5673 df-cnv 5674 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ec 8700 |
This theorem is referenced by: cossssid5 37797 dffunsALTV5 38013 |
Copyright terms: Public domain | W3C validator |