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Theorem fimacnv 5412
Description: The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
Assertion
Ref Expression
fimacnv (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)

Proof of Theorem fimacnv
StepHypRef Expression
1 imassrn 4772 . . 3 (𝐹𝐵) ⊆ ran 𝐹
2 dfdm4 4616 . . . 4 dom 𝐹 = ran 𝐹
3 fdm 5152 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
4 ssid 3042 . . . . 5 𝐴𝐴
53, 4syl6eqss 3074 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
62, 5syl5eqssr 3069 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐴)
71, 6syl5ss 3034 . 2 (𝐹:𝐴𝐵 → (𝐹𝐵) ⊆ 𝐴)
8 imassrn 4772 . . . 4 (𝐹𝐴) ⊆ ran 𝐹
9 frn 5155 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
108, 9syl5ss 3034 . . 3 (𝐹:𝐴𝐵 → (𝐹𝐴) ⊆ 𝐵)
11 ffun 5150 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
124, 3syl5sseqr 3073 . . . 4 (𝐹:𝐴𝐵𝐴 ⊆ dom 𝐹)
13 funimass3 5399 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
1411, 12, 13syl2anc 403 . . 3 (𝐹:𝐴𝐵 → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
1510, 14mpbid 145 . 2 (𝐹:𝐴𝐵𝐴 ⊆ (𝐹𝐵))
167, 15eqssd 3040 1 (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wss 2997  ccnv 4427  dom cdm 4428  ran crn 4429  cima 4431  Fun wfun 4996  wf 4998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010
This theorem is referenced by:  fmpt  5433  nn0supp  8695
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