Theorem feu 2767

Description: There is exactly one value of a function in its codomain.

Assertion

Ref	Expression
feu	⊢ ((F:A–→B ∧ C ∈ A) → ∃!y ∈ B ⟨C, y⟩ ∈ F)

Distinct variable group(s):   y,F   y,A   y,B   y,C

Proof of Theorem feu

Step	Hyp	Ref	Expression
1		fneu2 2729	. . . 4 ⊢ ((F Fn A ∧ C ∈ A) → ∃!y⟨C, y⟩ ∈ F)
2		ffn 2752	. . . 4 ⊢ (F:A–→B → F Fn A)
3	1, 2	sylan 343	. . 3 ⊢ ((F:A–→B ∧ C ∈ A) → ∃!y⟨C, y⟩ ∈ F)
4		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
5	4	opelf 2762	. . . . . . . 8 ⊢ ((F:A–→B ∧ ⟨C, y⟩ ∈ F) → (C ∈ A ∧ y ∈ B))
6	5	pm3.27d 262	. . . . . . 7 ⊢ ((F:A–→B ∧ ⟨C, y⟩ ∈ F) → y ∈ B)
7	6	exp 291	. . . . . 6 ⊢ (F:A–→B → (⟨C, y⟩ ∈ F → y ∈ B))
8		pm4.71r 482	. . . . . 6 ⊢ ((⟨C, y⟩ ∈ F → y ∈ B) ↔ (⟨C, y⟩ ∈ F ↔ (y ∈ B ∧ ⟨C, y⟩ ∈ F)))
9	7, 8	sylib 173	. . . . 5 ⊢ (F:A–→B → (⟨C, y⟩ ∈ F ↔ (y ∈ B ∧ ⟨C, y⟩ ∈ F)))
10	9	bieudv 1013	. . . 4 ⊢ (F:A–→B → (∃!y⟨C, y⟩ ∈ F ↔ ∃!y(y ∈ B ∧ ⟨C, y⟩ ∈ F)))
11	10	adantr 306	. . 3 ⊢ ((F:A–→B ∧ C ∈ A) → (∃!y⟨C, y⟩ ∈ F ↔ ∃!y(y ∈ B ∧ ⟨C, y⟩ ∈ F)))
12	3, 11	mpbid 170	. 2 ⊢ ((F:A–→B ∧ C ∈ A) → ∃!y(y ∈ B ∧ ⟨C, y⟩ ∈ F))
13		df-reu 1207	. 2 ⊢ (∃!y ∈ B ⟨C, y⟩ ∈ F ↔ ∃!y(y ∈ B ∧ ⟨C, y⟩ ∈ F))
14	12, 13	sylibr 175	1 ⊢ ((F:A–→B ∧ C ∈ A) → ∃!y ∈ B ⟨C, y⟩ ∈ F)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃!weu 1007   ∈ wcel 1092  ∃!wreu 1203  ⟨cop 1810   Fn wfn 2417  –→wf 2418

This theorem is referenced by:  fsn 2895

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-reu 1207  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434

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