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Theorem ineccnvmo2 36054
 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.)
Assertion
Ref Expression
ineccnvmo2 (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
Distinct variable group:   𝑢,𝐹,𝑥,𝑦

Proof of Theorem ineccnvmo2
StepHypRef Expression
1 ineccnvmo 36051 . 2 (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ↔ ∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥)
2 alrmomorn 36052 . 2 (∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥 ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
31, 2bitri 278 1 (∀𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]𝐹 ∩ [𝑦]𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 844  ∀wal 1536   = wceq 1538  ∃*wmo 2555  ∀wral 3070  ∃*wrmo 3073   ∩ cin 3857  ∅c0 4225   class class class wbr 5032  ◡ccnv 5523  ran crn 5525  [cec 8297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-rel 5531  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ec 8301 This theorem is referenced by:  cossssid5  36151  dffunsALTV5  36360
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