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Theorem eqfnfv2 5393
 Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 5-Feb-2004.)
Assertion
Ref Expression
eqfnfv2 ((F Fn A G Fn B) → (F = G ↔ (A = B x A (Fx) = (Gx))))
Distinct variable groups:   x,A   x,F   x,G
Allowed substitution hint:   B(x)

Proof of Theorem eqfnfv2
StepHypRef Expression
1 dmeq 4907 . . . 4 (F = G → dom F = dom G)
2 fndm 5182 . . . . 5 (F Fn A → dom F = A)
3 fndm 5182 . . . . 5 (G Fn B → dom G = B)
42, 3eqeqan12d 2368 . . . 4 ((F Fn A G Fn B) → (dom F = dom GA = B))
51, 4syl5ib 210 . . 3 ((F Fn A G Fn B) → (F = GA = B))
65pm4.71rd 616 . 2 ((F Fn A G Fn B) → (F = G ↔ (A = B F = G)))
7 fneq2 5174 . . . . . 6 (A = B → (G Fn AG Fn B))
87biimparc 473 . . . . 5 ((G Fn B A = B) → G Fn A)
9 eqfnfv 5392 . . . . 5 ((F Fn A G Fn A) → (F = Gx A (Fx) = (Gx)))
108, 9sylan2 460 . . . 4 ((F Fn A (G Fn B A = B)) → (F = Gx A (Fx) = (Gx)))
1110anassrs 629 . . 3 (((F Fn A G Fn B) A = B) → (F = Gx A (Fx) = (Gx)))
1211pm5.32da 622 . 2 ((F Fn A G Fn B) → ((A = B F = G) ↔ (A = B x A (Fx) = (Gx))))
136, 12bitrd 244 1 ((F Fn A G Fn B) → (F = G ↔ (A = B x A (Fx) = (Gx))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ∀wral 2614  dom cdm 4772   Fn wfn 4776   ‘cfv 4781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by:  eqfnfv3  5394  eqfunfv  5397  eqfnov  5589
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