Theorem fneu 2728

Description: There is exactly one value of a function.

Assertion

Ref	Expression
fneu	⊢ ((F Fn A ∧ B ∈ A) → ∃!y BFy)

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

Proof of Theorem fneu

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
2	1	anbi2d 468	. . . 4 ⊢ (x = B → ((F Fn A ∧ x ∈ A) ↔ (F Fn A ∧ B ∈ A)))
3		breq1 2065	. . . . 5 ⊢ (x = B → (xFy ↔ BFy))
4	3	bieudv 1013	. . . 4 ⊢ (x = B → (∃!y xFy ↔ ∃!y BFy))
5	2, 4	imbi12d 474	. . 3 ⊢ (x = B → (((F Fn A ∧ x ∈ A) → ∃!y xFy) ↔ ((F Fn A ∧ B ∈ A) → ∃!y BFy)))
6		fndm 2723	. . . . . . . 8 ⊢ (F Fn A → dom F = A)
7	6	eleq2d 1156	. . . . . . 7 ⊢ (F Fn A → (x ∈ dom F ↔ x ∈ A))
8		visset 1350	. . . . . . . 8 ⊢ x ∈ V
9	8	eldm 2527	. . . . . . 7 ⊢ (x ∈ dom F ↔ ∃y xFy)
10	7, 9	syl5bbr 412	. . . . . 6 ⊢ (F Fn A → (∃y xFy ↔ x ∈ A))
11		ax-17 925	. . . . . . . . 9 ⊢ (Fun F → ∀yFun F)
12		hbeu1 1015	. . . . . . . . 9 ⊢ (∃!y xFy → ∀y∃!y xFy)
13	11, 12	hbim 702	. . . . . . . 8 ⊢ ((Fun F → ∃!y xFy) → ∀y(Fun F → ∃!y xFy))
14		funeu 2685	. . . . . . . . . 10 ⊢ ((Fun F ∧ xFy) → ∃!y xFy)
15	14	exp 291	. . . . . . . . 9 ⊢ (Fun F → (xFy → ∃!y xFy))
16	15	com12 13	. . . . . . . 8 ⊢ (xFy → (Fun F → ∃!y xFy))
17	13, 16	19.23ai 746	. . . . . . 7 ⊢ (∃y xFy → (Fun F → ∃!y xFy))
18		fnfun 2721	. . . . . . 7 ⊢ (F Fn A → Fun F)
19	17, 18	syl5 22	. . . . . 6 ⊢ (∃y xFy → (F Fn A → ∃!y xFy))
20	10, 19	syl6bir 188	. . . . 5 ⊢ (F Fn A → (x ∈ A → (F Fn A → ∃!y xFy)))
21	20	pm2.43a 60	. . . 4 ⊢ (F Fn A → (x ∈ A → ∃!y xFy))
22	21	imp 277	. . 3 ⊢ ((F Fn A ∧ x ∈ A) → ∃!y xFy)
23	5, 22	vtoclg 1383	. 2 ⊢ (B ∈ A → ((F Fn A ∧ B ∈ A) → ∃!y BFy))
24	23	anabsi7 379	1 ⊢ ((F Fn A ∧ B ∈ A) → ∃!y BFy)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678  ∃!weu 1007   = wceq 1091   ∈ wcel 1092   class class class wbr 2054  dom cdm 2410  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  fneu2 2729  fnfvbr 2855

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-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-cnv 2426  df-co 2427  df-dm 2428  df-fun 2432  df-fn 2433

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