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Theorem dffo4 5423
 Description: Alternate definition of an onto mapping. (Contributed by set.mm contributors, 20-Mar-2007.)
Assertion
Ref Expression
dffo4 (F:AontoB ↔ (F:A–→B y B x A xFy))
Distinct variable groups:   x,y,A   x,B,y   x,F,y

Proof of Theorem dffo4
StepHypRef Expression
1 fof 5269 . . 3 (F:AontoBF:A–→B)
2 elrn 4896 . . . . . . . 8 (y ran Fx xFy)
3 forn 5272 . . . . . . . . 9 (F:AontoB → ran F = B)
43eleq2d 2420 . . . . . . . 8 (F:AontoB → (y ran Fy B))
52, 4syl5bbr 250 . . . . . . 7 (F:AontoB → (x xFyy B))
65biimpar 471 . . . . . 6 ((F:AontoB y B) → x xFy)
7 breldm 4911 . . . . . . . . . 10 (xFyx dom F)
8 fdm 5226 . . . . . . . . . . . 12 (F:A–→B → dom F = A)
91, 8syl 15 . . . . . . . . . . 11 (F:AontoB → dom F = A)
109eleq2d 2420 . . . . . . . . . 10 (F:AontoB → (x dom Fx A))
117, 10syl5ib 210 . . . . . . . . 9 (F:AontoB → (xFyx A))
1211ancrd 537 . . . . . . . 8 (F:AontoB → (xFy → (x A xFy)))
1312eximdv 1622 . . . . . . 7 (F:AontoB → (x xFyx(x A xFy)))
1413adantr 451 . . . . . 6 ((F:AontoB y B) → (x xFyx(x A xFy)))
156, 14mpd 14 . . . . 5 ((F:AontoB y B) → x(x A xFy))
16 df-rex 2620 . . . . 5 (x A xFyx(x A xFy))
1715, 16sylibr 203 . . . 4 ((F:AontoB y B) → x A xFy)
1817ralrimiva 2697 . . 3 (F:AontoBy B x A xFy)
191, 18jca 518 . 2 (F:AontoB → (F:A–→B y B x A xFy))
20 ffn 5223 . . . . . . 7 (F:A–→BF Fn A)
21 eqcom 2355 . . . . . . . . 9 (y = (Fx) ↔ (Fx) = y)
22 fnbrfvb 5358 . . . . . . . . 9 ((F Fn A x A) → ((Fx) = yxFy))
2321, 22syl5bb 248 . . . . . . . 8 ((F Fn A x A) → (y = (Fx) ↔ xFy))
2423biimprd 214 . . . . . . 7 ((F Fn A x A) → (xFyy = (Fx)))
2520, 24sylan 457 . . . . . 6 ((F:A–→B x A) → (xFyy = (Fx)))
2625reximdva 2726 . . . . 5 (F:A–→B → (x A xFyx A y = (Fx)))
2726ralimdv 2693 . . . 4 (F:A–→B → (y B x A xFyy B x A y = (Fx)))
2827imdistani 671 . . 3 ((F:A–→B y B x A xFy) → (F:A–→B y B x A y = (Fx)))
29 dffo3 5422 . . 3 (F:AontoB ↔ (F:A–→B y B x A y = (Fx)))
3028, 29sylibr 203 . 2 ((F:A–→B y B x A xFy) → F:AontoB)
3119, 30impbii 180 1 (F:AontoB ↔ (F:A–→B y B x A xFy))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   class class class wbr 4639  dom cdm 4772  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:  dffo5  5424
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