Theorem funbrfv 6548

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 6207	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 5457	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 572	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 4934	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 619	. . . . 5 ⊢ (𝑦 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑦) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2789	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) = 𝑦 ↔ (𝐹‘𝐴) = 𝐵))
7	5, 6	imbi12d 337	. . . 4 ⊢ (𝑦 = 𝐵 → (((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) ↔ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵)))
8		funeu 6215	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → ∃!𝑦 𝐴𝐹𝑦)
9		tz6.12-1 6523	. . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
10	8, 9	sylan2 583	. . . . 5 ⊢ ((𝐴𝐹𝑦 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑦)) → (𝐹‘𝐴) = 𝑦)
11	10	anabss7 660	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
12	7, 11	vtoclg 3486	. . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵))
13	3, 12	mpcom 38	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹‘𝐴) = 𝐵)
14	13	ex 405	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∃!weu 2582  Vcvv 3415   class class class wbr 4930  Rel wrel 5413  Fun wfun 6184  ‘cfv 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-iota 6154  df-fun 6192  df-fv 6198

This theorem is referenced by:  funopfv  6549  fnbrfvb  6550  fvelima  6563  fvi  6570  opabiota  6576  fmptco  6716  fliftfun  6890  fliftval  6894  tfrlem5  7822  fpwwe2  9865  nqerid  10155  sum0  14941  sumz  14942  fsumsers  14948  isumclim  14975  ntrivcvgfvn0  15118  ntrivcvgtail  15119  zprodn0  15156  iprodclim  15215  idinv  16920  cnextfvval  22380  cnextfres  22384  dvadd  24243  dvmul  24244  dvco  24250  dvcj  24253  dvrec  24258  dvcnv  24280  dvef  24283  ftc1cn  24346  ulmdv  24697  minvecolem4b  28436  minvecolem4  28438  hlimuni  28797  chscllem4  29201  fmptcof2  30167  fvtransport  33014  fvray  33123  fvline  33126  ftc1cnnc  34407  frege124d  39469  fvelimad  40946  fvelima2  40961