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Theorem imaiinfv 37776
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 6165 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fniinfv 6420 . . 3 ((𝐹𝐵) Fn 𝐵 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
31, 2syl 17 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 ((𝐹𝐵)‘𝑥) = ran (𝐹𝐵))
4 fvres 6369 . . . 4 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
54iineq2i 4692 . . 3 𝑥𝐵 ((𝐹𝐵)‘𝑥) = 𝑥𝐵 (𝐹𝑥)
65eqcomi 2769 . 2 𝑥𝐵 (𝐹𝑥) = 𝑥𝐵 ((𝐹𝐵)‘𝑥)
7 df-ima 5279 . . 3 (𝐹𝐵) = ran (𝐹𝐵)
87inteqi 4631 . 2 (𝐹𝐵) = ran (𝐹𝐵)
93, 6, 83eqtr4g 2819 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wss 3715   cint 4627   ciin 4673  ran crn 5267  cres 5268  cima 5269   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  elrfirn  37778
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