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Theorem fvelimab 5354
Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
Assertion
Ref Expression
fvelimab ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fvelimab
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2630 . . . 4 (𝐶 ∈ (𝐹𝐵) → 𝐶 ∈ V)
21anim2i 334 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ (𝐹𝐵)) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
3 ssel2 3020 . . . . . . . 8 ((𝐵𝐴𝑢𝐵) → 𝑢𝐴)
4 funfvex 5316 . . . . . . . . 9 ((Fun 𝐹𝑢 ∈ dom 𝐹) → (𝐹𝑢) ∈ V)
54funfni 5108 . . . . . . . 8 ((𝐹 Fn 𝐴𝑢𝐴) → (𝐹𝑢) ∈ V)
63, 5sylan2 280 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝐵𝐴𝑢𝐵)) → (𝐹𝑢) ∈ V)
76anassrs 392 . . . . . 6 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑢𝐵) → (𝐹𝑢) ∈ V)
8 eleq1 2150 . . . . . 6 ((𝐹𝑢) = 𝐶 → ((𝐹𝑢) ∈ V ↔ 𝐶 ∈ V))
97, 8syl5ibcom 153 . . . . 5 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑢𝐵) → ((𝐹𝑢) = 𝐶𝐶 ∈ V))
109rexlimdva 2489 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑢𝐵 (𝐹𝑢) = 𝐶𝐶 ∈ V))
1110imdistani 434 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ ∃𝑢𝐵 (𝐹𝑢) = 𝐶) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
12 eleq1 2150 . . . . . . 7 (𝑣 = 𝐶 → (𝑣 ∈ (𝐹𝐵) ↔ 𝐶 ∈ (𝐹𝐵)))
13 eqeq2 2097 . . . . . . . 8 (𝑣 = 𝐶 → ((𝐹𝑢) = 𝑣 ↔ (𝐹𝑢) = 𝐶))
1413rexbidv 2381 . . . . . . 7 (𝑣 = 𝐶 → (∃𝑢𝐵 (𝐹𝑢) = 𝑣 ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
1512, 14bibi12d 233 . . . . . 6 (𝑣 = 𝐶 → ((𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣) ↔ (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶)))
1615imbi2d 228 . . . . 5 (𝑣 = 𝐶 → (((𝐹 Fn 𝐴𝐵𝐴) → (𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))))
17 fnfun 5105 . . . . . . . 8 (𝐹 Fn 𝐴 → Fun 𝐹)
1817adantr 270 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → Fun 𝐹)
19 fndm 5107 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2019sseq2d 3054 . . . . . . . 8 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
2120biimpar 291 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
22 dfimafn 5347 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵) = {𝑣 ∣ ∃𝑢𝐵 (𝐹𝑢) = 𝑣})
2318, 21, 22syl2anc 403 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) = {𝑣 ∣ ∃𝑢𝐵 (𝐹𝑢) = 𝑣})
2423abeq2d 2200 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣))
2516, 24vtoclg 2679 . . . 4 (𝐶 ∈ V → ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶)))
2625impcom 123 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
272, 11, 26pm5.21nd 863 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
28 fveq2 5299 . . . 4 (𝑢 = 𝑥 → (𝐹𝑢) = (𝐹𝑥))
2928eqeq1d 2096 . . 3 (𝑢 = 𝑥 → ((𝐹𝑢) = 𝐶 ↔ (𝐹𝑥) = 𝐶))
3029cbvrexv 2591 . 2 (∃𝑢𝐵 (𝐹𝑢) = 𝐶 ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)
3127, 30syl6bb 194 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  {cab 2074  wrex 2360  Vcvv 2619  wss 2999  dom cdm 4436  cima 4439  Fun wfun 5004   Fn wfn 5005  cfv 5010
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 3955  ax-pow 4007  ax-pr 4034
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 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-fv 5018
This theorem is referenced by:  ssimaex  5359  foima2  5522  rexima  5526  ralima  5527  f1elima  5544  ovelimab  5787
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