MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqfnfv2 Structured version   Visualization version   GIF version

Theorem eqfnfv2 6502
Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
eqfnfv2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eqfnfv2
StepHypRef Expression
1 dmeq 5492 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
2 fndm 6168 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 fndm 6168 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3eqeqan12d 2781 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 = dom 𝐺𝐴 = 𝐵))
51, 4syl5ib 235 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺𝐴 = 𝐵))
65pm4.71rd 558 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵𝐹 = 𝐺)))
7 fneq2 6158 . . . . . 6 (𝐴 = 𝐵 → (𝐺 Fn 𝐴𝐺 Fn 𝐵))
87biimparc 471 . . . . 5 ((𝐺 Fn 𝐵𝐴 = 𝐵) → 𝐺 Fn 𝐴)
9 eqfnfv 6501 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
108, 9sylan2 586 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵𝐴 = 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
1110anassrs 459 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐴 = 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
1211pm5.32da 574 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴 = 𝐵𝐹 = 𝐺) ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
136, 12bitrd 270 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wral 3055  dom cdm 5277   Fn wfn 6063  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-fv 6076
This theorem is referenced by:  eqfnfv3  6503  eqfunfv  6506  eqfnov  6964  wfr3g  7616  2ffzeq  12668  eqwrd  13528  soseq  32130  frr3g  32155  sdclem2  33892  2ffzoeq  42004
  Copyright terms: Public domain W3C validator