Theorem fununmo 5372

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 5341	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦)
2		orc 719	. . . 4 ⊢ (𝑥𝐹𝑦 → (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦))
3		brun 4140	. . . 4 ⊢ (𝑥(𝐹 ∪ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦))
4	2, 3	sylibr 134	. . 3 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ∪ 𝐺)𝑦)
5	4	moimi 2145	. 2 ⊢ (∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦 → ∃*𝑦 𝑥𝐹𝑦)
6	1, 5	syl 14	1 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)

Syntax hints:   → wi 4   ∨ wo 715  ∃*wmo 2080   ∪ cun 3198   class class class wbr 4088  Fun wfun 5320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-fun 5328

This theorem is referenced by:  fununfun  5373