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Theorem resin 6871
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 6870 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
2 f1ofo 6856 . . . 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 6870 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐶𝐷)) → (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
53, 4syld3an3 1408 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
6 dfin4 4284 . . . 4 (𝐶𝐷) = (𝐶 ∖ (𝐶𝐷))
7 f1oeq3 6839 . . . 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 4284 . . . 4 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
10 f1oeq2 6838 . . . 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 5997 . . . 4 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴𝐵)))
13 f1oeq1 6837 . . . 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 1086   = wceq 1537  cdif 3960  cin 3962  ccnv 5688  cres 5691  Fun wfun 6557  ontowfo 6561  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by: (None)
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