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Theorem eqfnfv2 6205
Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
eqfnfv2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eqfnfv2
StepHypRef Expression
1 dmeq 5233 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
2 fndm 5890 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 fndm 5890 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3eqeqan12d 2625 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 = dom 𝐺𝐴 = 𝐵))
51, 4syl5ib 232 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺𝐴 = 𝐵))
65pm4.71rd 664 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵𝐹 = 𝐺)))
7 fneq2 5880 . . . . . 6 (𝐴 = 𝐵 → (𝐺 Fn 𝐴𝐺 Fn 𝐵))
87biimparc 502 . . . . 5 ((𝐺 Fn 𝐵𝐴 = 𝐵) → 𝐺 Fn 𝐴)
9 eqfnfv 6204 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
108, 9sylan2 489 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵𝐴 = 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
1110anassrs 677 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐴 = 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
1211pm5.32da 670 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴 = 𝐵𝐹 = 𝐺) ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
136, 12bitrd 266 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wral 2895  dom cdm 5028   Fn wfn 5785  cfv 5790
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 4712  ax-pow 4764  ax-pr 4828
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 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  eqfnfv3  6206  eqfunfv  6209  eqfnov  6642  wfr3g  7278  2ffzeq  12287  eqwrd  13150  soseq  30829  frr3g  30857  nodenselem4  30917  sdclem2  32532  2ffzoeq  40208
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