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Theorem resin 6807
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 6806 . . . 4 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
2 f1ofo 6792 . . . 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 6806 . . 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 4228 . . . 4 (𝐶𝐷) = (𝐶 ∖ (𝐶𝐷))
7 f1oeq3 6775 . . . 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 4228 . . . 4 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
10 f1oeq2 6774 . . . 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 5935 . . . 4 (𝐹 ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∖ (𝐴𝐵)))
13 f1oeq1 6773 . . . 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 3908  cin 3910  ccnv 5633  cres 5636  Fun wfun 6491  ontowfo 6495  1-1-ontowf1o 6496
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 5257  ax-nul 5264  ax-pr 5385
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 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by: (None)
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