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Theorem fvreseq 5196
Description: Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
fvreseq (((𝐹 Fn A 𝐺 Fn A) BA) → ((𝐹B) = (𝐺B) ↔ x B (𝐹x) = (𝐺x)))
Distinct variable groups:   x,B   x,𝐹   x,𝐺
Allowed substitution hint:   A(x)

Proof of Theorem fvreseq
StepHypRef Expression
1 fnssres 4938 . . . 4 ((𝐹 Fn A BA) → (𝐹B) Fn B)
2 fnssres 4938 . . . 4 ((𝐺 Fn A BA) → (𝐺B) Fn B)
31, 2anim12i 321 . . 3 (((𝐹 Fn A BA) (𝐺 Fn A BA)) → ((𝐹B) Fn B (𝐺B) Fn B))
43anandirs 514 . 2 (((𝐹 Fn A 𝐺 Fn A) BA) → ((𝐹B) Fn B (𝐺B) Fn B))
5 eqfnfv 5190 . . 3 (((𝐹B) Fn B (𝐺B) Fn B) → ((𝐹B) = (𝐺B) ↔ x B ((𝐹B)‘x) = ((𝐺B)‘x)))
6 fvres 5123 . . . . 5 (x B → ((𝐹B)‘x) = (𝐹x))
7 fvres 5123 . . . . 5 (x B → ((𝐺B)‘x) = (𝐺x))
86, 7eqeq12d 2036 . . . 4 (x B → (((𝐹B)‘x) = ((𝐺B)‘x) ↔ (𝐹x) = (𝐺x)))
98ralbiia 2316 . . 3 (x B ((𝐹B)‘x) = ((𝐺B)‘x) ↔ x B (𝐹x) = (𝐺x))
105, 9syl6bb 185 . 2 (((𝐹B) Fn B (𝐺B) Fn B) → ((𝐹B) = (𝐺B) ↔ x B (𝐹x) = (𝐺x)))
114, 10syl 14 1 (((𝐹 Fn A 𝐺 Fn A) BA) → ((𝐹B) = (𝐺B) ↔ x B (𝐹x) = (𝐺x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  wral 2284  wss 2894  cres 4274   Fn wfn 4824  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  tfri3  5875
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