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Theorem eqfnfvd 5400
Description: Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
eqfnfvd.1 (𝜑𝐹 Fn 𝐴)
eqfnfvd.2 (𝜑𝐺 Fn 𝐴)
eqfnfvd.3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
Assertion
Ref Expression
eqfnfvd (𝜑𝐹 = 𝐺)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥

Proof of Theorem eqfnfvd
StepHypRef Expression
1 eqfnfvd.3 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))
21ralrimiva 2446 . 2 (𝜑 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
3 eqfnfvd.1 . . 3 (𝜑𝐹 Fn 𝐴)
4 eqfnfvd.2 . . 3 (𝜑𝐺 Fn 𝐴)
5 eqfnfv 5397 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
63, 4, 5syl2anc 403 . 2 (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
72, 6mpbird 165 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359   Fn wfn 5010  cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  foeqcnvco  5569  f1eqcocnv  5570  tfrlem1  6073  frecrdg  6173  updjudhcoinlf  6769  updjudhcoinrg  6770  iseqvalt  9869  seq3val  9870  iseqoveq  9881  iseqsst  9882  iseqfeq2  9887  seq3feq2  9889  iseqfeq  9892  seq3shft  10268  efcvgfsum  10953  peano4nninf  11851  nninfalllemn  11853  nninfsellemeqinf  11863
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