Theorem fununmo 5325

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 5295	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦)
2		orc 714	. . . 4 ⊢ (𝑥𝐹𝑦 → (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦))
3		brun 4103	. . . 4 ⊢ (𝑥(𝐹 ∪ 𝐺)𝑦 ↔ (𝑥𝐹𝑦 ∨ 𝑥𝐺𝑦))
4	2, 3	sylibr 134	. . 3 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ∪ 𝐺)𝑦)
5	4	moimi 2120	. 2 ⊢ (∃*𝑦 𝑥(𝐹 ∪ 𝐺)𝑦 → ∃*𝑦 𝑥𝐹𝑦)
6	1, 5	syl 14	1 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)

Syntax hints:   → wi 4   ∨ wo 710  ∃*wmo 2056   ∪ cun 3168   class class class wbr 4051  Fun wfun 5274

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-fun 5282

This theorem is referenced by:  fununfun  5326