Theorem fneu 5187

Description: There is exactly one value of a function. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 22-Apr-2004.) (Revised by set.mm contributors, 18-Sep-2011.)

Assertion

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

Distinct variable groups:   y,F   y,B

Allowed substitution hint:   A(y)

Proof of Theorem fneu

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4642	. . . . . 6 ⊢ (x = B → (xFy ↔ BFy))
2	1	eubidv 2212	. . . . 5 ⊢ (x = B → (∃!y xFy ↔ ∃!y BFy))
3	2	imbi2d 307	. . . 4 ⊢ (x = B → ((Fun F → ∃!y xFy) ↔ (Fun F → ∃!y BFy)))
4		eldm 4898	. . . . 5 ⊢ (x ∈ dom F ↔ ∃y xFy)
5		funmo 5125	. . . . . 6 ⊢ (Fun F → ∃*y xFy)
6		exmoeu2 2247	. . . . . 6 ⊢ (∃y xFy → (∃*y xFy ↔ ∃!y xFy))
7	5, 6	syl5ib 210	. . . . 5 ⊢ (∃y xFy → (Fun F → ∃!y xFy))
8	4, 7	sylbi 187	. . . 4 ⊢ (x ∈ dom F → (Fun F → ∃!y xFy))
9	3, 8	vtoclga 2920	. . 3 ⊢ (B ∈ dom F → (Fun F → ∃!y BFy))
10	9	impcom 419	. 2 ⊢ ((Fun F ∧ B ∈ dom F) → ∃!y BFy)
11	10	funfni 5183	1 ⊢ ((F Fn A ∧ B ∈ A) → ∃!y BFy)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205   class class class wbr 4639  dom cdm 4772  Fun wfun 4775   Fn wfn 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790

This theorem is referenced by:  fneu2  5188  fnbrfvb  5358  mapsn  6026