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Theorem dffo3 5422
 Description: An onto mapping expressed in terms of function values. (Contributed by set.mm contributors, 29-Oct-2006.)
Assertion
Ref Expression
dffo3 (F:AontoB ↔ (F:A–→B y B x A y = (Fx)))
Distinct variable groups:   x,y,A   x,B,y   x,F,y

Proof of Theorem dffo3
StepHypRef Expression
1 dffo2 5273 . 2 (F:AontoB ↔ (F:A–→B ran F = B))
2 ffn 5223 . . . . 5 (F:A–→BF Fn A)
3 fnrnfv 5364 . . . . . 6 (F Fn A → ran F = {y x A y = (Fx)})
43eqeq1d 2361 . . . . 5 (F Fn A → (ran F = B ↔ {y x A y = (Fx)} = B))
52, 4syl 15 . . . 4 (F:A–→B → (ran F = B ↔ {y x A y = (Fx)} = B))
6 simpr 447 . . . . . . . . . . 11 (((F:A–→B x A) y = (Fx)) → y = (Fx))
7 ffvelrn 5415 . . . . . . . . . . . 12 ((F:A–→B x A) → (Fx) B)
87adantr 451 . . . . . . . . . . 11 (((F:A–→B x A) y = (Fx)) → (Fx) B)
96, 8eqeltrd 2427 . . . . . . . . . 10 (((F:A–→B x A) y = (Fx)) → y B)
109exp31 587 . . . . . . . . 9 (F:A–→B → (x A → (y = (Fx) → y B)))
1110rexlimdv 2737 . . . . . . . 8 (F:A–→B → (x A y = (Fx) → y B))
1211biantrurd 494 . . . . . . 7 (F:A–→B → ((y Bx A y = (Fx)) ↔ ((x A y = (Fx) → y B) (y Bx A y = (Fx)))))
13 dfbi2 609 . . . . . . 7 ((x A y = (Fx) ↔ y B) ↔ ((x A y = (Fx) → y B) (y Bx A y = (Fx))))
1412, 13syl6rbbr 255 . . . . . 6 (F:A–→B → ((x A y = (Fx) ↔ y B) ↔ (y Bx A y = (Fx))))
1514albidv 1625 . . . . 5 (F:A–→B → (y(x A y = (Fx) ↔ y B) ↔ y(y Bx A y = (Fx))))
16 abeq1 2459 . . . . 5 ({y x A y = (Fx)} = By(x A y = (Fx) ↔ y B))
17 df-ral 2619 . . . . 5 (y B x A y = (Fx) ↔ y(y Bx A y = (Fx)))
1815, 16, 173bitr4g 279 . . . 4 (F:A–→B → ({y x A y = (Fx)} = By B x A y = (Fx)))
195, 18bitrd 244 . . 3 (F:A–→B → (ran F = By B x A y = (Fx)))
2019pm5.32i 618 . 2 ((F:A–→B ran F = B) ↔ (F:A–→B y B x A y = (Fx)))
211, 20bitri 240 1 (F:AontoB ↔ (F:A–→B y B x A y = (Fx)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  ∃wrex 2615  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-f 4791  df-fo 4793  df-fv 4795 This theorem is referenced by:  dffo4  5423  foelrn  5425  foco2  5426  foov  5606
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