Theorem funeu 2685

Description: There is exactly one value of a function.

Assertion

Ref	Expression
funeu	⊢ ((Fun F ∧ xFy) → ∃!y xFy)

Distinct variable group(s):   x,y,F

Proof of Theorem funeu

Step	Hyp	Ref	Expression
1		19.8a 712	. . . 4 ⊢ (xFy → ∃y xFy)
2		dffun3 2675	. . . . . 6 ⊢ (Fun F ↔ (Rel F ∧ ∀x∃z∀y(xFy → y = z)))
3	2	pm3.27bd 263	. . . . 5 ⊢ (Fun F → ∀x∃z∀y(xFy → y = z))
4	3	19.21bi 742	. . . 4 ⊢ (Fun F → ∃z∀y(xFy → y = z))
5	1, 4	anim12i 268	. . 3 ⊢ ((xFy ∧ Fun F) → (∃y xFy ∧ ∃z∀y(xFy → y = z)))
6		ax-17 925	. . . 4 ⊢ (xFy → ∀z xFy)
7	6	eu3 1024	. . 3 ⊢ (∃!y xFy ↔ (∃y xFy ∧ ∃z∀y(xFy → y = z)))
8	5, 7	sylibr 175	. 2 ⊢ ((xFy ∧ Fun F) → ∃!y xFy)
9	8	ancoms 334	1 ⊢ ((Fun F ∧ xFy) → ∃!y xFy)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007   class class class wbr 2054  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  funeu2 2686  fneu 2728  funbrfv 2852  fvelrn 2883

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-fun 2432

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