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Theorem eqfnfv2f 5415
 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 5411 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 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (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 5411 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
2 eqfnfv2f.1 . . . . 5 𝑥𝐹
3 nfcv 2229 . . . . 5 𝑥𝑧
42, 3nffv 5328 . . . 4 𝑥(𝐹𝑧)
5 eqfnfv2f.2 . . . . 5 𝑥𝐺
65, 3nffv 5328 . . . 4 𝑥(𝐺𝑧)
74, 6nfeq 2237 . . 3 𝑥(𝐹𝑧) = (𝐺𝑧)
8 nfv 1467 . . 3 𝑧(𝐹𝑥) = (𝐺𝑥)
9 fveq2 5318 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
10 fveq2 5318 . . . 4 (𝑧 = 𝑥 → (𝐺𝑧) = (𝐺𝑥))
119, 10eqeq12d 2103 . . 3 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑥) = (𝐺𝑥)))
127, 8, 11cbvral 2587 . 2 (∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
131, 12syl6bb 195 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290  Ⅎwnfc 2216  ∀wral 2360   Fn wfn 5023  ‘cfv 5028 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036 This theorem is referenced by: (None)
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