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Theorem eqfnfv2 5209
 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 A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B x A (𝐹x) = (𝐺x))))
Distinct variable groups:   x,A   x,𝐹   x,𝐺
Allowed substitution hint:   B(x)

Proof of Theorem eqfnfv2
StepHypRef Expression
1 dmeq 4478 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
2 fndm 4941 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
3 fndm 4941 . . . . 5 (𝐺 Fn B → dom 𝐺 = B)
42, 3eqeqan12d 2052 . . . 4 ((𝐹 Fn A 𝐺 Fn B) → (dom 𝐹 = dom 𝐺A = B))
51, 4syl5ib 143 . . 3 ((𝐹 Fn A 𝐺 Fn B) → (𝐹 = 𝐺A = B))
65pm4.71rd 374 . 2 ((𝐹 Fn A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B 𝐹 = 𝐺)))
7 fneq2 4931 . . . . . 6 (A = B → (𝐺 Fn A𝐺 Fn B))
87biimparc 283 . . . . 5 ((𝐺 Fn B A = B) → 𝐺 Fn A)
9 eqfnfv 5208 . . . . 5 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
108, 9sylan2 270 . . . 4 ((𝐹 Fn A (𝐺 Fn B A = B)) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
1110anassrs 380 . . 3 (((𝐹 Fn A 𝐺 Fn B) A = B) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
1211pm5.32da 425 . 2 ((𝐹 Fn A 𝐺 Fn B) → ((A = B 𝐹 = 𝐺) ↔ (A = B x A (𝐹x) = (𝐺x))))
136, 12bitrd 177 1 ((𝐹 Fn A 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B x A (𝐹x) = (𝐺x))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∀wral 2300  dom cdm 4288   Fn wfn 4840  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853 This theorem is referenced by:  eqfnfv3  5210  eqfunfv  5213  eqfnov  5549  2ffzeq  8768
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