Theorem ineccnvmo2 39437

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 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)

Distinct variable group:   𝑢,𝐹,𝑥,𝑦

Proof of Theorem ineccnvmo2

Step	Hyp	Ref	Expression
1		ineccnvmo 39426	. 2 ⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥)
2		alrmomorn 39427	. 2 ⊢ (∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥 ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
3	1, 2	bitri 278	1 ⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861  ∀wal 1567   = wceq 1569  ∃*wmo 2562  ∀wral 3076  ∃*wrmo 3364   ∩ cin 4088  ∅c0 4277   class class class wbr 5102  ^◡ccnv 5650  ran crn 5652  [cec 8723

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8727

This theorem is used by:  cossssid5  39630  dffunsALTV5  39841