Theorem fununmo 6568

Description: If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)

Assertion

Ref	Expression
fununmo	⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐹   𝑦,𝐺

Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem fununmo

Step	Hyp	Ref	Expression
1		funmo 6537	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦)
2		orc 878	. . . 4 ⊢ (𝑥𝐹𝑦 → (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦))
3		brun 5151	. . . 4 ⊢ (𝑥(𝐹 ∪ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦))
4	2, 3	sylibr 236	. . 3 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ∪ 𝐺)𝑦)
5	4	moimi 2572	. 2 ⊢ (∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦 → ∃*𝑦 𝑥𝐹𝑦)
6	1, 5	syl 17	1 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)

Syntax hints:   → wi 4   ∨ wo 858  ∃*wmo 2564   ∪ cun 3902   class class class wbr 5100  Fun wfun 6515

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-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-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-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-fun 6523

This theorem is referenced by:  fununfun  6569