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Theorem eqfnfv2 5519
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 4739 . . . 4 (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺)
2 fndm 5222 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3 fndm 5222 . . . . 5 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
42, 3eqeqan12d 2155 . . . 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 5212 . . . . . 6 (𝐴 = 𝐵 → (𝐺 Fn 𝐴𝐺 Fn 𝐵))
87biimparc 297 . . . . 5 ((𝐺 Fn 𝐵𝐴 = 𝐵) → 𝐺 Fn 𝐴)
9 eqfnfv 5518 . . . . 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 1331  wral 2416  dom cdm 4539   Fn wfn 5118  cfv 5123
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by:  eqfnfv3  5520  eqfunfv  5523  eqfnov  5877  2ffzeq  9925
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