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Theorem elpreima 5229
Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
elpreima (𝐹 Fn A → (B (𝐹𝐶) ↔ (B A (𝐹B) 𝐶)))

Proof of Theorem elpreima
StepHypRef Expression
1 cnvimass 4631 . . . . 5 (𝐹𝐶) ⊆ dom 𝐹
21sseli 2935 . . . 4 (B (𝐹𝐶) → B dom 𝐹)
3 fndm 4941 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
43eleq2d 2104 . . . 4 (𝐹 Fn A → (B dom 𝐹B A))
52, 4syl5ib 143 . . 3 (𝐹 Fn A → (B (𝐹𝐶) → B A))
6 fnfun 4939 . . . . 5 (𝐹 Fn A → Fun 𝐹)
7 fvimacnvi 5224 . . . . 5 ((Fun 𝐹 B (𝐹𝐶)) → (𝐹B) 𝐶)
86, 7sylan 267 . . . 4 ((𝐹 Fn A B (𝐹𝐶)) → (𝐹B) 𝐶)
98ex 108 . . 3 (𝐹 Fn A → (B (𝐹𝐶) → (𝐹B) 𝐶))
105, 9jcad 291 . 2 (𝐹 Fn A → (B (𝐹𝐶) → (B A (𝐹B) 𝐶)))
11 fvimacnv 5225 . . . . 5 ((Fun 𝐹 B dom 𝐹) → ((𝐹B) 𝐶B (𝐹𝐶)))
1211funfni 4942 . . . 4 ((𝐹 Fn A B A) → ((𝐹B) 𝐶B (𝐹𝐶)))
1312biimpd 132 . . 3 ((𝐹 Fn A B A) → ((𝐹B) 𝐶B (𝐹𝐶)))
1413expimpd 345 . 2 (𝐹 Fn A → ((B A (𝐹B) 𝐶) → B (𝐹𝐶)))
1510, 14impbid 120 1 (𝐹 Fn A → (B (𝐹𝐶) ↔ (B A (𝐹B) 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  ccnv 4287  dom cdm 4288  cima 4291  Fun wfun 4839   Fn wfn 4840  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fniniseg  5230  fncnvima2  5231  rexsupp  5234  unpreima  5235  respreima  5238
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