Theorem funbrfv 6912

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 6536	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 5695	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 580	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 5114	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 630	. . . . 5 ⊢ (𝑦 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑦) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2742	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) = 𝑦 ↔ (𝐹‘𝐴) = 𝐵))
7	5, 6	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝐵 → (((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) ↔ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵)))
8		funeu 6544	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9		tz6.12-1 6884	. . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
10	8, 9	sylan2 593	. . . . 5 ⊢ ((𝐴𝐹𝑦 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑦)) → (𝐹‘𝐴) = 𝑦)
11	10	anabss7 673	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
12	7, 11	vtoclg 3523	. . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵))
13	3, 12	mpcom 38	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵)
14	13	ex 412	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!weu 2562  Vcvv 3450   class class class wbr 5110  Rel wrel 5646  Fun wfun 6508  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522

This theorem is referenced by:  funopfv  6913  fnbrfvb  6914  fvelima2  6916  fvelima  6929  fvelimad  6931  fvi  6940  opabiota  6946  fmptco  7104  fliftfun  7290  fliftval  7294  tfrlem5  8351  fpwwe2  10603  nqerid  10893  sum0  15694  sumz  15695  fsumsers  15701  isumclim  15730  ntrivcvgfvn0  15872  ntrivcvgtail  15873  zprodn0  15912  iprodclim  15971  idinv  17758  cnextfvval  23959  cnextfres  23963  dvadd  25850  dvmul  25851  dvco  25858  dvcj  25861  dvrec  25866  dvcnv  25888  dvef  25891  ftc1cn  25957  ulmdv  26319  minvecolem4b  30814  minvecolem4  30816  hlimuni  31174  chscllem4  31576  fmptcof2  32588  fvtransport  36027  fvray  36136  fvline  36139  ftc1cnnc  37693  iscard4  43529  frege124d  43757