Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imaiinfv Structured version   Visualization version   GIF version

Theorem imaiinfv 36274
 Description: Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
imaiinfv ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem imaiinfv
StepHypRef Expression
1 fnssres 5918 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fniinfv 6167 . . 3 ((𝐹𝐵) Fn 𝐵 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
31, 2syl 17 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
4 fvres 6117 . . . 4 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
54iineq2i 4476 . . 3 𝑥𝐵 ((𝐹𝐵)‘𝑥) = 𝑥𝐵 (𝐹𝑥)
65eqcomi 2619 . 2 𝑥𝐵 (𝐹𝑥) = 𝑥𝐵 ((𝐹𝐵)‘𝑥)
7 df-ima 5051 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
87inteqi 4414 . 2 (𝐹𝐵) = ran (𝐹𝐵)
93, 6, 83eqtr4g 2669 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ⊆ wss 3540  ∩ cint 4410  ∩ ciin 4456  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  elrfirn  36276
 Copyright terms: Public domain W3C validator