Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dff1o6 Structured version   GIF version

Theorem dff1o6 5359
 Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
Assertion
Ref Expression
dff1o6 (𝐹:A1-1-ontoB ↔ (𝐹 Fn A ran 𝐹 = B x A y A ((𝐹x) = (𝐹y) → x = y)))
Distinct variable groups:   x,y,A   x,𝐹,y
Allowed substitution hints:   B(x,y)

Proof of Theorem dff1o6
StepHypRef Expression
1 df-f1o 4852 . 2 (𝐹:A1-1-ontoB ↔ (𝐹:A1-1B 𝐹:AontoB))
2 dff13 5350 . . 3 (𝐹:A1-1B ↔ (𝐹:AB x A y A ((𝐹x) = (𝐹y) → x = y)))
3 df-fo 4851 . . 3 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
42, 3anbi12i 433 . 2 ((𝐹:A1-1B 𝐹:AontoB) ↔ ((𝐹:AB x A y A ((𝐹x) = (𝐹y) → x = y)) (𝐹 Fn A ran 𝐹 = B)))
5 df-3an 886 . . 3 ((𝐹 Fn A ran 𝐹 = B x A y A ((𝐹x) = (𝐹y) → x = y)) ↔ ((𝐹 Fn A ran 𝐹 = B) x A y A ((𝐹x) = (𝐹y) → x = y)))
6 eqimss 2991 . . . . . . 7 (ran 𝐹 = B → ran 𝐹B)
76anim2i 324 . . . . . 6 ((𝐹 Fn A ran 𝐹 = B) → (𝐹 Fn A ran 𝐹B))
8 df-f 4849 . . . . . 6 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
97, 8sylibr 137 . . . . 5 ((𝐹 Fn A ran 𝐹 = B) → 𝐹:AB)
109pm4.71ri 372 . . . 4 ((𝐹 Fn A ran 𝐹 = B) ↔ (𝐹:AB (𝐹 Fn A ran 𝐹 = B)))
1110anbi1i 431 . . 3 (((𝐹 Fn A ran 𝐹 = B) x A y A ((𝐹x) = (𝐹y) → x = y)) ↔ ((𝐹:AB (𝐹 Fn A ran 𝐹 = B)) x A y A ((𝐹x) = (𝐹y) → x = y)))
12 an32 496 . . 3 (((𝐹:AB (𝐹 Fn A ran 𝐹 = B)) x A y A ((𝐹x) = (𝐹y) → x = y)) ↔ ((𝐹:AB x A y A ((𝐹x) = (𝐹y) → x = y)) (𝐹 Fn A ran 𝐹 = B)))
135, 11, 123bitrri 196 . 2 (((𝐹:AB x A y A ((𝐹x) = (𝐹y) → x = y)) (𝐹 Fn A ran 𝐹 = B)) ↔ (𝐹 Fn A ran 𝐹 = B x A y A ((𝐹x) = (𝐹y) → x = y)))
141, 4, 133bitri 195 1 (𝐹:A1-1-ontoB ↔ (𝐹 Fn A ran 𝐹 = B x A y A ((𝐹x) = (𝐹y) → x = y)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242  ∀wral 2300   ⊆ wss 2911  ran crn 4289   Fn wfn 4840  ⟶wf 4841  –1-1→wf1 4842  –onto→wfo 4843  –1-1-onto→wf1o 4844  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator