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Theorem fcofo 5367
Description: An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
fcofo ((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) → 𝐹:AontoB)

Proof of Theorem fcofo
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 903 . 2 ((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) → 𝐹:AB)
2 ffvelrn 5243 . . . . 5 ((𝑆:BA y B) → (𝑆y) A)
323ad2antl2 1066 . . . 4 (((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) y B) → (𝑆y) A)
4 simpl3 908 . . . . . 6 (((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) y B) → (𝐹𝑆) = ( I ↾ B))
54fveq1d 5123 . . . . 5 (((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) y B) → ((𝐹𝑆)‘y) = (( I ↾ B)‘y))
6 fvco3 5187 . . . . . 6 ((𝑆:BA y B) → ((𝐹𝑆)‘y) = (𝐹‘(𝑆y)))
763ad2antl2 1066 . . . . 5 (((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) y B) → ((𝐹𝑆)‘y) = (𝐹‘(𝑆y)))
8 fvresi 5299 . . . . . 6 (y B → (( I ↾ B)‘y) = y)
98adantl 262 . . . . 5 (((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) y B) → (( I ↾ B)‘y) = y)
105, 7, 93eqtr3rd 2078 . . . 4 (((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) y B) → y = (𝐹‘(𝑆y)))
11 fveq2 5121 . . . . . 6 (x = (𝑆y) → (𝐹x) = (𝐹‘(𝑆y)))
1211eqeq2d 2048 . . . . 5 (x = (𝑆y) → (y = (𝐹x) ↔ y = (𝐹‘(𝑆y))))
1312rspcev 2650 . . . 4 (((𝑆y) A y = (𝐹‘(𝑆y))) → x A y = (𝐹x))
143, 10, 13syl2anc 391 . . 3 (((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) y B) → x A y = (𝐹x))
1514ralrimiva 2386 . 2 ((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) → y B x A y = (𝐹x))
16 dffo3 5257 . 2 (𝐹:AontoB ↔ (𝐹:AB y B x A y = (𝐹x)))
171, 15, 16sylanbrc 394 1 ((𝐹:AB 𝑆:BA (𝐹𝑆) = ( I ↾ B)) → 𝐹:AontoB)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wral 2300  wrex 2301   I cid 4016  cres 4290  ccom 4292  wf 4841  ontowfo 4843  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-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853
This theorem is referenced by:  fcof1o  5372
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