Theorem funbrfv 6870

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 6498	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 5668	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 580	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 5093	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 630	. . . . 5 ⊢ (𝑦 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑦) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2743	. . . . 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 6506	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9		tz6.12-1 6845	. . . . . 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 3507	. . 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 1541   ∈ wcel 2111  ∃!weu 2563  Vcvv 3436   class class class wbr 5089  Rel wrel 5619  Fun wfun 6475  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489

This theorem is referenced by:  funopfv  6871  fnbrfvb  6872  fvelima2  6874  fvelima  6887  fvelimad  6889  fvi  6898  opabiota  6904  fmptco  7062  fliftfun  7246  fliftval  7250  tfrlem5  8299  fpwwe2  10534  nqerid  10824  sum0  15628  sumz  15629  fsumsers  15635  isumclim  15664  ntrivcvgfvn0  15806  ntrivcvgtail  15807  zprodn0  15846  iprodclim  15905  idinv  17696  cnextfvval  23980  cnextfres  23984  dvadd  25870  dvmul  25871  dvco  25878  dvcj  25881  dvrec  25886  dvcnv  25908  dvef  25911  ftc1cn  25977  ulmdv  26339  minvecolem4b  30858  minvecolem4  30860  hlimuni  31218  chscllem4  31620  fmptcof2  32639  fvtransport  36076  fvray  36185  fvline  36188  ftc1cnnc  37742  iscard4  43636  frege124d  43864