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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xppreima2 Structured version   Visualization version   GIF version

Theorem xppreima2 31613
Description: The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
Hypotheses
Ref Expression
xppreima2.1 (šœ‘ ā†’ š¹:š“āŸ¶šµ)
xppreima2.2 (šœ‘ ā†’ šŗ:š“āŸ¶š¶)
xppreima2.3 š» = (š‘„ āˆˆ š“ ā†¦ āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ©)
Assertion
Ref Expression
xppreima2 (šœ‘ ā†’ (ā—”š» ā€œ (š‘Œ Ɨ š‘)) = ((ā—”š¹ ā€œ š‘Œ) āˆ© (ā—”šŗ ā€œ š‘)))
Distinct variable groups:   š‘„,š“   š‘„,šµ   š‘„,š¶   š‘„,š¹   š‘„,šŗ   š‘„,š»   šœ‘,š‘„
Allowed substitution hints:   š‘Œ(š‘„)   š‘(š‘„)

Proof of Theorem xppreima2
StepHypRef Expression
1 xppreima2.3 . . . 4 š» = (š‘„ āˆˆ š“ ā†¦ āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ©)
21funmpt2 6541 . . 3 Fun š»
3 xppreima2.1 . . . . . . . 8 (šœ‘ ā†’ š¹:š“āŸ¶šµ)
43ffvelcdmda 7036 . . . . . . 7 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (š¹ā€˜š‘„) āˆˆ šµ)
5 xppreima2.2 . . . . . . . 8 (šœ‘ ā†’ šŗ:š“āŸ¶š¶)
65ffvelcdmda 7036 . . . . . . 7 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (šŗā€˜š‘„) āˆˆ š¶)
7 opelxp 5670 . . . . . . 7 (āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ© āˆˆ (šµ Ɨ š¶) ā†” ((š¹ā€˜š‘„) āˆˆ šµ āˆ§ (šŗā€˜š‘„) āˆˆ š¶))
84, 6, 7sylanbrc 584 . . . . . 6 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ© āˆˆ (šµ Ɨ š¶))
98, 1fmptd 7063 . . . . 5 (šœ‘ ā†’ š»:š“āŸ¶(šµ Ɨ š¶))
109frnd 6677 . . . 4 (šœ‘ ā†’ ran š» āŠ† (šµ Ɨ š¶))
11 xpss 5650 . . . 4 (šµ Ɨ š¶) āŠ† (V Ɨ V)
1210, 11sstrdi 3957 . . 3 (šœ‘ ā†’ ran š» āŠ† (V Ɨ V))
13 xppreima 31608 . . 3 ((Fun š» āˆ§ ran š» āŠ† (V Ɨ V)) ā†’ (ā—”š» ā€œ (š‘Œ Ɨ š‘)) = ((ā—”(1st āˆ˜ š») ā€œ š‘Œ) āˆ© (ā—”(2nd āˆ˜ š») ā€œ š‘)))
142, 12, 13sylancr 588 . 2 (šœ‘ ā†’ (ā—”š» ā€œ (š‘Œ Ɨ š‘)) = ((ā—”(1st āˆ˜ š») ā€œ š‘Œ) āˆ© (ā—”(2nd āˆ˜ š») ā€œ š‘)))
15 fo1st 7942 . . . . . . . . 9 1st :Vā€“ontoā†’V
16 fofn 6759 . . . . . . . . 9 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
1715, 16ax-mp 5 . . . . . . . 8 1st Fn V
18 opex 5422 . . . . . . . . 9 āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ© āˆˆ V
1918, 1fnmpti 6645 . . . . . . . 8 š» Fn š“
20 ssv 3969 . . . . . . . 8 ran š» āŠ† V
21 fnco 6619 . . . . . . . 8 ((1st Fn V āˆ§ š» Fn š“ āˆ§ ran š» āŠ† V) ā†’ (1st āˆ˜ š») Fn š“)
2217, 19, 20, 21mp3an 1462 . . . . . . 7 (1st āˆ˜ š») Fn š“
2322a1i 11 . . . . . 6 (šœ‘ ā†’ (1st āˆ˜ š») Fn š“)
243ffnd 6670 . . . . . 6 (šœ‘ ā†’ š¹ Fn š“)
252a1i 11 . . . . . . . . . 10 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ Fun š»)
2612adantr 482 . . . . . . . . . 10 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ ran š» āŠ† (V Ɨ V))
27 simpr 486 . . . . . . . . . . 11 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘„ āˆˆ š“)
2818, 1dmmpti 6646 . . . . . . . . . . 11 dom š» = š“
2927, 28eleqtrrdi 2845 . . . . . . . . . 10 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘„ āˆˆ dom š»)
30 opfv 31607 . . . . . . . . . 10 (((Fun š» āˆ§ ran š» āŠ† (V Ɨ V)) āˆ§ š‘„ āˆˆ dom š») ā†’ (š»ā€˜š‘„) = āŸØ((1st āˆ˜ š»)ā€˜š‘„), ((2nd āˆ˜ š»)ā€˜š‘„)āŸ©)
3125, 26, 29, 30syl21anc 837 . . . . . . . . 9 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (š»ā€˜š‘„) = āŸØ((1st āˆ˜ š»)ā€˜š‘„), ((2nd āˆ˜ š»)ā€˜š‘„)āŸ©)
321fvmpt2 6960 . . . . . . . . . 10 ((š‘„ āˆˆ š“ āˆ§ āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ© āˆˆ (šµ Ɨ š¶)) ā†’ (š»ā€˜š‘„) = āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ©)
3327, 8, 32syl2anc 585 . . . . . . . . 9 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (š»ā€˜š‘„) = āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ©)
3431, 33eqtr3d 2775 . . . . . . . 8 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ āŸØ((1st āˆ˜ š»)ā€˜š‘„), ((2nd āˆ˜ š»)ā€˜š‘„)āŸ© = āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ©)
35 fvex 6856 . . . . . . . . 9 ((1st āˆ˜ š»)ā€˜š‘„) āˆˆ V
36 fvex 6856 . . . . . . . . 9 ((2nd āˆ˜ š»)ā€˜š‘„) āˆˆ V
3735, 36opth 5434 . . . . . . . 8 (āŸØ((1st āˆ˜ š»)ā€˜š‘„), ((2nd āˆ˜ š»)ā€˜š‘„)āŸ© = āŸØ(š¹ā€˜š‘„), (šŗā€˜š‘„)āŸ© ā†” (((1st āˆ˜ š»)ā€˜š‘„) = (š¹ā€˜š‘„) āˆ§ ((2nd āˆ˜ š»)ā€˜š‘„) = (šŗā€˜š‘„)))
3834, 37sylib 217 . . . . . . 7 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (((1st āˆ˜ š»)ā€˜š‘„) = (š¹ā€˜š‘„) āˆ§ ((2nd āˆ˜ š»)ā€˜š‘„) = (šŗā€˜š‘„)))
3938simpld 496 . . . . . 6 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ ((1st āˆ˜ š»)ā€˜š‘„) = (š¹ā€˜š‘„))
4023, 24, 39eqfnfvd 6986 . . . . 5 (šœ‘ ā†’ (1st āˆ˜ š») = š¹)
4140cnveqd 5832 . . . 4 (šœ‘ ā†’ ā—”(1st āˆ˜ š») = ā—”š¹)
4241imaeq1d 6013 . . 3 (šœ‘ ā†’ (ā—”(1st āˆ˜ š») ā€œ š‘Œ) = (ā—”š¹ ā€œ š‘Œ))
43 fo2nd 7943 . . . . . . . . 9 2nd :Vā€“ontoā†’V
44 fofn 6759 . . . . . . . . 9 (2nd :Vā€“ontoā†’V ā†’ 2nd Fn V)
4543, 44ax-mp 5 . . . . . . . 8 2nd Fn V
46 fnco 6619 . . . . . . . 8 ((2nd Fn V āˆ§ š» Fn š“ āˆ§ ran š» āŠ† V) ā†’ (2nd āˆ˜ š») Fn š“)
4745, 19, 20, 46mp3an 1462 . . . . . . 7 (2nd āˆ˜ š») Fn š“
4847a1i 11 . . . . . 6 (šœ‘ ā†’ (2nd āˆ˜ š») Fn š“)
495ffnd 6670 . . . . . 6 (šœ‘ ā†’ šŗ Fn š“)
5038simprd 497 . . . . . 6 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ ((2nd āˆ˜ š»)ā€˜š‘„) = (šŗā€˜š‘„))
5148, 49, 50eqfnfvd 6986 . . . . 5 (šœ‘ ā†’ (2nd āˆ˜ š») = šŗ)
5251cnveqd 5832 . . . 4 (šœ‘ ā†’ ā—”(2nd āˆ˜ š») = ā—”šŗ)
5352imaeq1d 6013 . . 3 (šœ‘ ā†’ (ā—”(2nd āˆ˜ š») ā€œ š‘) = (ā—”šŗ ā€œ š‘))
5442, 53ineq12d 4174 . 2 (šœ‘ ā†’ ((ā—”(1st āˆ˜ š») ā€œ š‘Œ) āˆ© (ā—”(2nd āˆ˜ š») ā€œ š‘)) = ((ā—”š¹ ā€œ š‘Œ) āˆ© (ā—”šŗ ā€œ š‘)))
5514, 54eqtrd 2773 1 (šœ‘ ā†’ (ā—”š» ā€œ (š‘Œ Ɨ š‘)) = ((ā—”š¹ ā€œ š‘Œ) āˆ© (ā—”šŗ ā€œ š‘)))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   āˆ§ wa 397   = wceq 1542   āˆˆ wcel 2107  Vcvv 3444   āˆ© cin 3910   āŠ† wss 3911  āŸØcop 4593   ā†¦ cmpt 5189   Ɨ cxp 5632  ā—”ccnv 5633  dom cdm 5634  ran crn 5635   ā€œ cima 5637   āˆ˜ ccom 5638  Fun wfun 6491   Fn wfn 6492  āŸ¶wf 6493  ā€“ontoā†’wfo 6495  ā€˜cfv 6497  1st c1st 7920  2nd c2nd 7921
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-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  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-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  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-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  mbfmco2  32922
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