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Theorem eqfnfv2f 5212
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 5208 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 5208 . 2 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺z A (𝐹z) = (𝐺z)))
2 eqfnfv2f.1 . . . . 5 x𝐹
3 nfcv 2175 . . . . 5 xz
42, 3nffv 5128 . . . 4 x(𝐹z)
5 eqfnfv2f.2 . . . . 5 x𝐺
65, 3nffv 5128 . . . 4 x(𝐺z)
74, 6nfeq 2182 . . 3 x(𝐹z) = (𝐺z)
8 nfv 1418 . . 3 z(𝐹x) = (𝐺x)
9 fveq2 5121 . . . 4 (z = x → (𝐹z) = (𝐹x))
10 fveq2 5121 . . . 4 (z = x → (𝐺z) = (𝐺x))
119, 10eqeq12d 2051 . . 3 (z = x → ((𝐹z) = (𝐺z) ↔ (𝐹x) = (𝐺x)))
127, 8, 11cbvral 2523 . 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 1242  wnfc 2162  wral 2300   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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