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Theorem resin 6856
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 6855 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
2 f1ofo 6841 . . . 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 6855 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐶𝐷)) → (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
53, 4syld3an3 1410 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴 ∖ (𝐴𝐵))):(𝐴 ∖ (𝐴𝐵))–1-1-onto→(𝐶 ∖ (𝐶𝐷)))
6 dfin4 4268 . . . 4 (𝐶𝐷) = (𝐶 ∖ (𝐶𝐷))
7 f1oeq3 6824 . . . 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 4268 . . . 4 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
10 f1oeq2 6823 . . . 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 5979 . . . 4 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴𝐵)))
13 f1oeq1 6822 . . . 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 217 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 205  w3a 1088   = wceq 1542  cdif 3946  cin 3948  ccnv 5676  cres 5679  Fun wfun 6538  ontowfo 6542  1-1-ontowf1o 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551
This theorem is referenced by: (None)
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