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Theorem imaiinfv 42681
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 6692 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fniinfv 6987 . . 3 ((𝐹𝐵) Fn 𝐵 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
31, 2syl 17 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
4 fvres 6926 . . . 4 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
54iineq2i 5019 . . 3 𝑥𝐵 ((𝐹𝐵)‘𝑥) = 𝑥𝐵 (𝐹𝑥)
65eqcomi 2744 . 2 𝑥𝐵 (𝐹𝑥) = 𝑥𝐵 ((𝐹𝐵)‘𝑥)
7 df-ima 5702 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
87inteqi 4955 . 2 (𝐹𝐵) = ran (𝐹𝐵)
93, 6, 83eqtr4g 2800 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wss 3963   cint 4951   ciin 4997  ran crn 5690  cres 5691  cima 5692   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  elrfirn  42683
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