Theorem ineccnvmo2 38349

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 38346	. 2 ⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 ∈ ran 𝐹 𝑢𝐹𝑥)
2		alrmomorn 38347	. 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)

Syntax hints:   ↔ wb 206   ∨ wo 847  ∀wal 1538   = wceq 1540  ∃*wmo 2532  ∀wral 3045  ∃*wrmo 3355   ∩ cin 3916  ∅c0 4299   class class class wbr 5110  ^◡ccnv 5640  ran crn 5642  [cec 8672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rmo 3356  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ec 8676

This theorem is referenced by:  cossssid5  38469  dffunsALTV5  38686