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Theorem eqfnfv2f 5297
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 5293 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 5293 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
2 eqfnfv2f.1 . . . . 5 𝑥𝐹
3 nfcv 2194 . . . . 5 𝑥𝑧
42, 3nffv 5213 . . . 4 𝑥(𝐹𝑧)
5 eqfnfv2f.2 . . . . 5 𝑥𝐺
65, 3nffv 5213 . . . 4 𝑥(𝐺𝑧)
74, 6nfeq 2201 . . 3 𝑥(𝐹𝑧) = (𝐺𝑧)
8 nfv 1437 . . 3 𝑧(𝐹𝑥) = (𝐺𝑥)
9 fveq2 5206 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
10 fveq2 5206 . . . 4 (𝑧 = 𝑥 → (𝐺𝑧) = (𝐺𝑥))
119, 10eqeq12d 2070 . . 3 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑥) = (𝐺𝑥)))
127, 8, 11cbvral 2546 . 2 (∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
131, 12syl6bb 189 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnfc 2181  wral 2323   Fn wfn 4925  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by: (None)
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