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Theorem imaiinfv 42709
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 6690 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fniinfv 6986 . . 3 ((𝐹𝐵) Fn 𝐵 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
31, 2syl 17 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
4 fvres 6924 . . . 4 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
54iineq2i 5013 . . 3 𝑥𝐵 ((𝐹𝐵)‘𝑥) = 𝑥𝐵 (𝐹𝑥)
65eqcomi 2745 . 2 𝑥𝐵 (𝐹𝑥) = 𝑥𝐵 ((𝐹𝐵)‘𝑥)
7 df-ima 5697 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
87inteqi 4949 . 2 (𝐹𝐵) = ran (𝐹𝐵)
93, 6, 83eqtr4g 2801 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wss 3950   cint 4945   ciin 4991  ran crn 5685  cres 5686  cima 5687   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by:  elrfirn  42711
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