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Theorem fvelimab 5256
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 2583 . . . 4 (𝐶 ∈ (𝐹𝐵) → 𝐶 ∈ V)
21anim2i 328 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ (𝐹𝐵)) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
3 ssel2 2967 . . . . . . . 8 ((𝐵𝐴𝑢𝐵) → 𝑢𝐴)
4 funfvex 5219 . . . . . . . . 9 ((Fun 𝐹𝑢 ∈ dom 𝐹) → (𝐹𝑢) ∈ V)
54funfni 5026 . . . . . . . 8 ((𝐹 Fn 𝐴𝑢𝐴) → (𝐹𝑢) ∈ V)
63, 5sylan2 274 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝐵𝐴𝑢𝐵)) → (𝐹𝑢) ∈ V)
76anassrs 386 . . . . . 6 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑢𝐵) → (𝐹𝑢) ∈ V)
8 eleq1 2116 . . . . . 6 ((𝐹𝑢) = 𝐶 → ((𝐹𝑢) ∈ V ↔ 𝐶 ∈ V))
97, 8syl5ibcom 148 . . . . 5 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑢𝐵) → ((𝐹𝑢) = 𝐶𝐶 ∈ V))
109rexlimdva 2450 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑢𝐵 (𝐹𝑢) = 𝐶𝐶 ∈ V))
1110imdistani 427 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ ∃𝑢𝐵 (𝐹𝑢) = 𝐶) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
12 eleq1 2116 . . . . . . 7 (𝑣 = 𝐶 → (𝑣 ∈ (𝐹𝐵) ↔ 𝐶 ∈ (𝐹𝐵)))
13 eqeq2 2065 . . . . . . . 8 (𝑣 = 𝐶 → ((𝐹𝑢) = 𝑣 ↔ (𝐹𝑢) = 𝐶))
1413rexbidv 2344 . . . . . . 7 (𝑣 = 𝐶 → (∃𝑢𝐵 (𝐹𝑢) = 𝑣 ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
1512, 14bibi12d 228 . . . . . 6 (𝑣 = 𝐶 → ((𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣) ↔ (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶)))
1615imbi2d 223 . . . . 5 (𝑣 = 𝐶 → (((𝐹 Fn 𝐴𝐵𝐴) → (𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))))
17 fnfun 5023 . . . . . . . 8 (𝐹 Fn 𝐴 → Fun 𝐹)
1817adantr 265 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → Fun 𝐹)
19 fndm 5025 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2019sseq2d 3000 . . . . . . . 8 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
2120biimpar 285 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
22 dfimafn 5249 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵) = {𝑣 ∣ ∃𝑢𝐵 (𝐹𝑢) = 𝑣})
2318, 21, 22syl2anc 397 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) = {𝑣 ∣ ∃𝑢𝐵 (𝐹𝑢) = 𝑣})
2423abeq2d 2166 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣))
2516, 24vtoclg 2630 . . . 4 (𝐶 ∈ V → ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶)))
2625impcom 120 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
272, 11, 26pm5.21nd 836 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
28 fveq2 5205 . . . 4 (𝑢 = 𝑥 → (𝐹𝑢) = (𝐹𝑥))
2928eqeq1d 2064 . . 3 (𝑢 = 𝑥 → ((𝐹𝑢) = 𝐶 ↔ (𝐹𝑥) = 𝐶))
3029cbvrexv 2551 . 2 (∃𝑢𝐵 (𝐹𝑢) = 𝐶 ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)
3127, 30syl6bb 189 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  {cab 2042  wrex 2324  Vcvv 2574  wss 2944  dom cdm 4372  cima 4375  Fun wfun 4923   Fn wfn 4924  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937
This theorem is referenced by:  ssimaex  5261  rexima  5421  ralima  5422  f1elima  5439  ovelimab  5678
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