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Theorem eqfnfv2f 6207
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 6203 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 6203 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
2 eqfnfv2f.1 . . . . 5 𝑥𝐹
3 nfcv 2750 . . . . 5 𝑥𝑧
42, 3nffv 6094 . . . 4 𝑥(𝐹𝑧)
5 eqfnfv2f.2 . . . . 5 𝑥𝐺
65, 3nffv 6094 . . . 4 𝑥(𝐺𝑧)
74, 6nfeq 2761 . . 3 𝑥(𝐹𝑧) = (𝐺𝑧)
8 nfv 1829 . . 3 𝑧(𝐹𝑥) = (𝐺𝑥)
9 fveq2 6087 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
10 fveq2 6087 . . . 4 (𝑧 = 𝑥 → (𝐺𝑧) = (𝐺𝑥))
119, 10eqeq12d 2624 . . 3 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑥) = (𝐺𝑥)))
127, 8, 11cbvral 3142 . 2 (∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
131, 12syl6bb 274 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wnfc 2737  wral 2895   Fn wfn 5784  cfv 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-fv 5797
This theorem is referenced by:  aacllem  42298
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