Theorem fununiq 5518

Description: Implicational form of part of the definition of a function. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
fununiq	⊢ ((Fun F ∧ AFB ∧ AFC) → B = C)

Proof of Theorem fununiq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . . . 5 ⊢ (AFB → (A ∈ V ∧ B ∈ V))
2		brex 4690	. . . . 5 ⊢ (AFC → (A ∈ V ∧ C ∈ V))
3	1, 2	anim12i 549	. . . 4 ⊢ ((AFB ∧ AFC) → ((A ∈ V ∧ B ∈ V) ∧ (A ∈ V ∧ C ∈ V)))
4		anandi 801	. . . 4 ⊢ ((A ∈ V ∧ (B ∈ V ∧ C ∈ V)) ↔ ((A ∈ V ∧ B ∈ V) ∧ (A ∈ V ∧ C ∈ V)))
5	3, 4	sylibr 203	. . 3 ⊢ ((AFB ∧ AFC) → (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
6	5	3adant1 973	. 2 ⊢ ((Fun F ∧ AFB ∧ AFC) → (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
7		dffun2 5120	. . . . . 6 ⊢ (Fun F ↔ ∀x∀y∀z((xFy ∧ xFz) → y = z))
8		breq12 4645	. . . . . . . . . 10 ⊢ ((x = A ∧ y = B) → (xFy ↔ AFB))
9	8	3adant3 975	. . . . . . . . 9 ⊢ ((x = A ∧ y = B ∧ z = C) → (xFy ↔ AFB))
10		breq12 4645	. . . . . . . . . 10 ⊢ ((x = A ∧ z = C) → (xFz ↔ AFC))
11	10	3adant2 974	. . . . . . . . 9 ⊢ ((x = A ∧ y = B ∧ z = C) → (xFz ↔ AFC))
12	9, 11	anbi12d 691	. . . . . . . 8 ⊢ ((x = A ∧ y = B ∧ z = C) → ((xFy ∧ xFz) ↔ (AFB ∧ AFC)))
13		eqeq12 2365	. . . . . . . . 9 ⊢ ((y = B ∧ z = C) → (y = z ↔ B = C))
14	13	3adant1 973	. . . . . . . 8 ⊢ ((x = A ∧ y = B ∧ z = C) → (y = z ↔ B = C))
15	12, 14	imbi12d 311	. . . . . . 7 ⊢ ((x = A ∧ y = B ∧ z = C) → (((xFy ∧ xFz) → y = z) ↔ ((AFB ∧ AFC) → B = C)))
16	15	spc3gv 2945	. . . . . 6 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (∀x∀y∀z((xFy ∧ xFz) → y = z) → ((AFB ∧ AFC) → B = C)))
17	7, 16	syl5bi 208	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (Fun F → ((AFB ∧ AFC) → B = C)))
18	17	exp4a 589	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (Fun F → (AFB → (AFC → B = C))))
19	18	3impd 1165	. . 3 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → ((Fun F ∧ AFB ∧ AFC) → B = C))
20	19	3expb 1152	. 2 ⊢ ((A ∈ V ∧ (B ∈ V ∧ C ∈ V)) → ((Fun F ∧ AFB ∧ AFC) → B = C))
21	6, 20	mpcom 32	1 ⊢ ((Fun F ∧ AFB ∧ AFC) → B = C)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2860   class class class wbr 4640  Fun wfun 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-id 4768  df-cnv 4786  df-fun 4790

This theorem is referenced by:  funsi  5521  fntxp  5805  fnpprod  5844  enpw1  6063  enprmaplem3  6079