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Theorem fcof1 5366
Description: An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
fcof1 ((𝐹:AB (𝑅𝐹) = ( I ↾ A)) → 𝐹:A1-1B)

Proof of Theorem fcof1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 102 . 2 ((𝐹:AB (𝑅𝐹) = ( I ↾ A)) → 𝐹:AB)
2 simprr 484 . . . . . . . 8 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → (𝐹x) = (𝐹y))
32fveq2d 5125 . . . . . . 7 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → (𝑅‘(𝐹x)) = (𝑅‘(𝐹y)))
4 simpll 481 . . . . . . . 8 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → 𝐹:AB)
5 simprll 489 . . . . . . . 8 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → x A)
6 fvco3 5187 . . . . . . . 8 ((𝐹:AB x A) → ((𝑅𝐹)‘x) = (𝑅‘(𝐹x)))
74, 5, 6syl2anc 391 . . . . . . 7 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → ((𝑅𝐹)‘x) = (𝑅‘(𝐹x)))
8 simprlr 490 . . . . . . . 8 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → y A)
9 fvco3 5187 . . . . . . . 8 ((𝐹:AB y A) → ((𝑅𝐹)‘y) = (𝑅‘(𝐹y)))
104, 8, 9syl2anc 391 . . . . . . 7 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → ((𝑅𝐹)‘y) = (𝑅‘(𝐹y)))
113, 7, 103eqtr4d 2079 . . . . . 6 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → ((𝑅𝐹)‘x) = ((𝑅𝐹)‘y))
12 simplr 482 . . . . . . 7 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → (𝑅𝐹) = ( I ↾ A))
1312fveq1d 5123 . . . . . 6 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → ((𝑅𝐹)‘x) = (( I ↾ A)‘x))
1412fveq1d 5123 . . . . . 6 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → ((𝑅𝐹)‘y) = (( I ↾ A)‘y))
1511, 13, 143eqtr3d 2077 . . . . 5 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → (( I ↾ A)‘x) = (( I ↾ A)‘y))
16 fvresi 5299 . . . . . 6 (x A → (( I ↾ A)‘x) = x)
175, 16syl 14 . . . . 5 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → (( I ↾ A)‘x) = x)
18 fvresi 5299 . . . . . 6 (y A → (( I ↾ A)‘y) = y)
198, 18syl 14 . . . . 5 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → (( I ↾ A)‘y) = y)
2015, 17, 193eqtr3d 2077 . . . 4 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) ((x A y A) (𝐹x) = (𝐹y))) → x = y)
2120expr 357 . . 3 (((𝐹:AB (𝑅𝐹) = ( I ↾ A)) (x A y A)) → ((𝐹x) = (𝐹y) → x = y))
2221ralrimivva 2395 . 2 ((𝐹:AB (𝑅𝐹) = ( I ↾ A)) → x A y A ((𝐹x) = (𝐹y) → x = y))
23 dff13 5350 . 2 (𝐹:A1-1B ↔ (𝐹:AB x A y A ((𝐹x) = (𝐹y) → x = y)))
241, 22, 23sylanbrc 394 1 ((𝐹:AB (𝑅𝐹) = ( I ↾ A)) → 𝐹:A1-1B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wral 2300   I cid 4016  cres 4290  ccom 4292  wf 4841  1-1wf1 4842  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-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fv 4853
This theorem is referenced by:  fcof1o  5372
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