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Theorem eqfnfv2f 5194
Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5190 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
Hypotheses
Ref Expression
eqfnfv2f.1 x𝐹
eqfnfv2f.2 x𝐺
Assertion
Ref Expression
eqfnfv2f ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
Distinct variable group:   x,A
Allowed substitution hints:   𝐹(x)   𝐺(x)

Proof of Theorem eqfnfv2f
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eqfnfv 5190 . 2 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺z A (𝐹z) = (𝐺z)))
2 eqfnfv2f.1 . . . . 5 x𝐹
3 nfcv 2160 . . . . 5 xz
42, 3nffv 5110 . . . 4 x(𝐹z)
5 eqfnfv2f.2 . . . . 5 x𝐺
65, 3nffv 5110 . . . 4 x(𝐺z)
74, 6nfeq 2167 . . 3 x(𝐹z) = (𝐺z)
8 nfv 1402 . . 3 z(𝐹x) = (𝐺x)
9 fveq2 5103 . . . 4 (z = x → (𝐹z) = (𝐹x))
10 fveq2 5103 . . . 4 (z = x → (𝐺z) = (𝐺x))
119, 10eqeq12d 2036 . . 3 (z = x → ((𝐹z) = (𝐺z) ↔ (𝐹x) = (𝐺x)))
127, 8, 11cbvral 2507 . 2 (z A (𝐹z) = (𝐺z) ↔ x A (𝐹x) = (𝐺x))
131, 12syl6bb 185 1 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wnfc 2147  wral 2284   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-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by: (None)
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