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Theorem fununiq 5517
 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.
StepHypRef Expression
1 brex 4689 . . . . 5 (AFB → (A V B V))
2 brex 4689 . . . . 5 (AFC → (A V C V))
31, 2anim12i 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)))
53, 4sylibr 203 . . 3 ((AFB AFC) → (A V (B V C V)))
653adant1 973 . 2 ((Fun F AFB AFC) → (A V (B V C V)))
7 dffun2 5119 . . . . . 6 (Fun Fxyz((xFy xFz) → y = z))
8 breq12 4644 . . . . . . . . . 10 ((x = A y = B) → (xFyAFB))
983adant3 975 . . . . . . . . 9 ((x = A y = B z = C) → (xFyAFB))
10 breq12 4644 . . . . . . . . . 10 ((x = A z = C) → (xFzAFC))
11103adant2 974 . . . . . . . . 9 ((x = A y = B z = C) → (xFzAFC))
129, 11anbi12d 691 . . . . . . . 8 ((x = A y = B z = C) → ((xFy xFz) ↔ (AFB AFC)))
13 eqeq12 2365 . . . . . . . . 9 ((y = B z = C) → (y = zB = C))
14133adant1 973 . . . . . . . 8 ((x = A y = B z = C) → (y = zB = C))
1512, 14imbi12d 311 . . . . . . 7 ((x = A y = B z = C) → (((xFy xFz) → y = z) ↔ ((AFB AFC) → B = C)))
1615spc3gv 2944 . . . . . 6 ((A V B V C V) → (xyz((xFy xFz) → y = z) → ((AFB AFC) → B = C)))
177, 16syl5bi 208 . . . . 5 ((A V B V C V) → (Fun F → ((AFB AFC) → B = C)))
1817exp4a 589 . . . 4 ((A V B V C V) → (Fun F → (AFB → (AFCB = C))))
19183impd 1165 . . 3 ((A V B V C V) → ((Fun F AFB AFC) → B = C))
20193expb 1152 . 2 ((A V (B V C V)) → ((Fun F AFB AFC) → B = C))
216, 20mpcom 32 1 ((Fun F AFB AFC) → B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859   class class class wbr 4639  Fun wfun 4775 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-id 4767  df-cnv 4785  df-fun 4789 This theorem is referenced by:  funsi  5520  fntxp  5804  fnpprod  5843  enpw1  6062  enprmaplem3  6078
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