Users' Mathboxes 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 36140
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 5803 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fniinfv 6050 . . 3 ((𝐹𝐵) Fn 𝐵 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
31, 2syl 17 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
4 fvres 6000 . . . 4 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
54iineq2i 4374 . . 3 𝑥𝐵 ((𝐹𝐵)‘𝑥) = 𝑥𝐵 (𝐹𝑥)
65eqcomi 2523 . 2 𝑥𝐵 (𝐹𝑥) = 𝑥𝐵 ((𝐹𝐵)‘𝑥)
7 df-ima 4945 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
87inteqi 4312 . 2 (𝐹𝐵) = ran (𝐹𝐵)
93, 6, 83eqtr4g 2573 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wss 3444   cint 4308   ciin 4354  ran crn 4933  cres 4934  cima 4935   Fn wfn 5684  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-int 4309  df-iin 4356  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-fv 5697
This theorem is referenced by:  elrfirn  36142
  Copyright terms: Public domain W3C validator