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Theorem eqfnfv2 5487
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 4709 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
2 fndm 5192 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 fndm 5192 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3eqeqan12d 2133 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 = dom 𝐺𝐴 = 𝐵))
51, 4syl5ib 153 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺𝐴 = 𝐵))
65pm4.71rd 391 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵𝐹 = 𝐺)))
7 fneq2 5182 . . . . . 6 (𝐴 = 𝐵 → (𝐺 Fn 𝐴𝐺 Fn 𝐵))
87biimparc 297 . . . . 5 ((𝐺 Fn 𝐵𝐴 = 𝐵) → 𝐺 Fn 𝐴)
9 eqfnfv 5486 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
108, 9sylan2 284 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵𝐴 = 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
1110anassrs 397 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐴 = 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
1211pm5.32da 447 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴 = 𝐵𝐹 = 𝐺) ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
136, 12bitrd 187 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wral 2393  dom cdm 4509   Fn wfn 5088  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  eqfnfv3  5488  eqfunfv  5491  eqfnov  5845  2ffzeq  9886
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