Theorem ineccnvmo2 38867

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 38856	. 2 ⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥)
2		alrmomorn 38857	. 2 ⊢ (∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥 ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
3	1, 2	bitri 277	1 ⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)

Syntax hints:   ↔ wb 208   ∨ wo 858  ∀wal 1558   = wceq 1560  ∃*wmo 2564  ∀wral 3076  ∃*wrmo 3366   ∩ cin 3903  ∅c0 4285   class class class wbr 5100  ^◡ccnv 5646  ran crn 5648  [cec 8676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680

This theorem is referenced by:  cossssid5  39060  dffunsALTV5  39271