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Theorem resin 6870
Description: The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
Assertion
Ref Expression
resin ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))

Proof of Theorem resin
StepHypRef Expression
1 resdif 6869 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
2 f1ofo 6855 . . . 4 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐶𝐷))
31, 2syl 17 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐶𝐷))
4 resdif 6869 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐶𝐷)) → (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
53, 4syld3an3 1411 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
6 dfin4 4278 . . . 4 (𝐶𝐷) = (𝐶 ∖ (𝐶𝐷))
7 f1oeq3 6838 . . . 4 ((𝐶𝐷) = (𝐶 ∖ (𝐶𝐷)) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶 ∖ (𝐶𝐷))))
86, 7ax-mp 5 . . 3 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
9 dfin4 4278 . . . 4 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
10 f1oeq2 6837 . . . 4 ((𝐴𝐵) = (𝐴 ∖ (𝐴𝐵)) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷))))
119, 10ax-mp 5 . . 3 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
129reseq2i 5994 . . . 4 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴𝐵)))
13 f1oeq1 6836 . . . 4 ((𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))) → ((𝐹 ↾ (𝐴𝐵)):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷))))
1412, 13ax-mp 5 . . 3 ((𝐹 ↾ (𝐴𝐵)):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
158, 11, 143bitrri 298 . 2 ((𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
165, 15sylib 218 1 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  cdif 3948  cin 3950  ccnv 5684  cres 5687  Fun wfun 6555  ontowfo 6559  1-1-ontowf1o 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by: (None)
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