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Theorem eqfnfv2 7052
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 5914 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
2 fndm 6671 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 fndm 6671 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3eqeqan12d 2751 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 = dom 𝐺𝐴 = 𝐵))
51, 4imbitrid 244 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺𝐴 = 𝐵))
65pm4.71rd 562 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵𝐹 = 𝐺)))
7 fneq2 6660 . . . . . 6 (𝐴 = 𝐵 → (𝐺 Fn 𝐴𝐺 Fn 𝐵))
87biimparc 479 . . . . 5 ((𝐺 Fn 𝐵𝐴 = 𝐵) → 𝐺 Fn 𝐴)
9 eqfnfv 7051 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
108, 9sylan2 593 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵𝐴 = 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
1110anassrs 467 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐴 = 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
1211pm5.32da 579 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴 = 𝐵𝐹 = 𝐺) ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
136, 12bitrd 279 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wral 3061  dom cdm 5685   Fn wfn 6556  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  eqfnfv3  7053  eqfunfv  7056  eqfnov  7562  soseq  8184  fpr3g  8310  wfr3g  8347  frr3g  9796  2ffzeq  13689  eqwrd  14595  sdclem2  37749  eqfnfv2d2  41982  aks6d1c5  42140  2ffzoeq  47339
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