Theorem funbrfv 5616

Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
funbrfv	⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))

Proof of Theorem funbrfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 5287	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 4715	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 283	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 4047	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 464	. . . . 5 ⊢ (𝑦 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑦) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2214	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) = 𝑦 ↔ (𝐹‘𝐴) = 𝐵))
7	5, 6	imbi12d 234	. . . 4 ⊢ (𝑦 = 𝐵 → (((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) ↔ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵)))
8		funeu 5295	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9		tz6.12-1 5602	. . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
10	8, 9	sylan2 286	. . . . 5 ⊢ ((𝐴𝐹𝑦 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑦)) → (𝐹‘𝐴) = 𝑦)
11	10	anabss7 583	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
12	7, 11	vtoclg 2832	. . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵))
13	3, 12	mpcom 36	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵)
14	13	ex 115	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372  ∃!weu 2053   ∈ wcel 2175  Vcvv 2771   class class class wbr 4043  Rel wrel 4679  Fun wfun 5264  ‘cfv 5270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278

This theorem is referenced by:  funopfv  5617  fnbrfvb  5618  fvelima  5629  fvi  5635  fmptco  5745  fliftfun  5864  fliftval  5868  tfrlem5  6399  sum0  11641  isumz  11642  fsumsersdc  11648  isumclim  11674  zprodap0  11834  dvaddxx  15117  dvmulxx  15118  dvcj  15123  dvrecap  15127  dvef  15141  pilem3  15197