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Theorem dff1o6 5341
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 4836 . 2 (𝐹:A1-1-ontoB ↔ (𝐹:A1-1B 𝐹:AontoB))
2 dff13 5332 . . 3 (𝐹:A1-1B ↔ (𝐹:AB x A y A ((𝐹x) = (𝐹y) → x = y)))
3 df-fo 4835 . . 3 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
42, 3anbi12i 436 . 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 875 . . 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 2974 . . . . . . 7 (ran 𝐹 = B → ran 𝐹B)
76anim2i 324 . . . . . 6 ((𝐹 Fn A ran 𝐹 = B) → (𝐹 Fn A ran 𝐹B))
8 df-f 4833 . . . . . 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 434 . . 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 484 . . 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 873   = wceq 1228  wral 2284  wss 2894  ran crn 4273   Fn wfn 4824  wf 4825  1-1wf1 4826  ontowfo 4827  1-1-ontowf1o 4828  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837
This theorem is referenced by: (None)
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