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Theorem eqfnfv2f 5256
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 5252 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 5252 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
2 eqfnfv2f.1 . . . . 5 𝑥𝐹
3 nfcv 2178 . . . . 5 𝑥𝑧
42, 3nffv 5172 . . . 4 𝑥(𝐹𝑧)
5 eqfnfv2f.2 . . . . 5 𝑥𝐺
65, 3nffv 5172 . . . 4 𝑥(𝐺𝑧)
74, 6nfeq 2185 . . 3 𝑥(𝐹𝑧) = (𝐺𝑧)
8 nfv 1421 . . 3 𝑧(𝐹𝑥) = (𝐺𝑥)
9 fveq2 5165 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
10 fveq2 5165 . . . 4 (𝑧 = 𝑥 → (𝐺𝑧) = (𝐺𝑥))
119, 10eqeq12d 2054 . . 3 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑥) = (𝐺𝑥)))
127, 8, 11cbvral 2526 . 2 (∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
131, 12syl6bb 185 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wnfc 2165  wral 2303   Fn wfn 4884  cfv 4889
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-mpt 3817  df-id 4027  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-iota 4854  df-fun 4891  df-fn 4892  df-fv 4897
This theorem is referenced by: (None)
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