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Theorem fvelimab 5154
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 A BA) → (𝐶 (𝐹B) ↔ x B (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 2543 . . . 4 (𝐶 (𝐹B) → 𝐶 V)
21anim2i 324 . . 3 (((𝐹 Fn A BA) 𝐶 (𝐹B)) → ((𝐹 Fn A BA) 𝐶 V))
3 ssel2 2917 . . . . . . . 8 ((BA u B) → u A)
4 funfvex 5117 . . . . . . . . 9 ((Fun 𝐹 u dom 𝐹) → (𝐹u) V)
54funfni 4925 . . . . . . . 8 ((𝐹 Fn A u A) → (𝐹u) V)
63, 5sylan2 270 . . . . . . 7 ((𝐹 Fn A (BA u B)) → (𝐹u) V)
76anassrs 382 . . . . . 6 (((𝐹 Fn A BA) u B) → (𝐹u) V)
8 eleq1 2082 . . . . . 6 ((𝐹u) = 𝐶 → ((𝐹u) V ↔ 𝐶 V))
97, 8syl5ibcom 144 . . . . 5 (((𝐹 Fn A BA) u B) → ((𝐹u) = 𝐶𝐶 V))
109rexlimdva 2411 . . . 4 ((𝐹 Fn A BA) → (u B (𝐹u) = 𝐶𝐶 V))
1110imdistani 422 . . 3 (((𝐹 Fn A BA) u B (𝐹u) = 𝐶) → ((𝐹 Fn A BA) 𝐶 V))
12 eleq1 2082 . . . . . . 7 (v = 𝐶 → (v (𝐹B) ↔ 𝐶 (𝐹B)))
13 eqeq2 2031 . . . . . . . 8 (v = 𝐶 → ((𝐹u) = v ↔ (𝐹u) = 𝐶))
1413rexbidv 2305 . . . . . . 7 (v = 𝐶 → (u B (𝐹u) = vu B (𝐹u) = 𝐶))
1512, 14bibi12d 224 . . . . . 6 (v = 𝐶 → ((v (𝐹B) ↔ u B (𝐹u) = v) ↔ (𝐶 (𝐹B) ↔ u B (𝐹u) = 𝐶)))
1615imbi2d 219 . . . . 5 (v = 𝐶 → (((𝐹 Fn A BA) → (v (𝐹B) ↔ u B (𝐹u) = v)) ↔ ((𝐹 Fn A BA) → (𝐶 (𝐹B) ↔ u B (𝐹u) = 𝐶))))
17 fnfun 4922 . . . . . . . 8 (𝐹 Fn A → Fun 𝐹)
1817adantr 261 . . . . . . 7 ((𝐹 Fn A BA) → Fun 𝐹)
19 fndm 4924 . . . . . . . . 9 (𝐹 Fn A → dom 𝐹 = A)
2019sseq2d 2950 . . . . . . . 8 (𝐹 Fn A → (B ⊆ dom 𝐹BA))
2120biimpar 281 . . . . . . 7 ((𝐹 Fn A BA) → B ⊆ dom 𝐹)
22 dfimafn 5147 . . . . . . 7 ((Fun 𝐹 B ⊆ dom 𝐹) → (𝐹B) = {vu B (𝐹u) = v})
2318, 21, 22syl2anc 393 . . . . . 6 ((𝐹 Fn A BA) → (𝐹B) = {vu B (𝐹u) = v})
2423abeq2d 2132 . . . . 5 ((𝐹 Fn A BA) → (v (𝐹B) ↔ u B (𝐹u) = v))
2516, 24vtoclg 2590 . . . 4 (𝐶 V → ((𝐹 Fn A BA) → (𝐶 (𝐹B) ↔ u B (𝐹u) = 𝐶)))
2625impcom 116 . . 3 (((𝐹 Fn A BA) 𝐶 V) → (𝐶 (𝐹B) ↔ u B (𝐹u) = 𝐶))
272, 11, 26pm5.21nd 815 . 2 ((𝐹 Fn A BA) → (𝐶 (𝐹B) ↔ u B (𝐹u) = 𝐶))
28 fveq2 5103 . . . 4 (u = x → (𝐹u) = (𝐹x))
2928eqeq1d 2030 . . 3 (u = x → ((𝐹u) = 𝐶 ↔ (𝐹x) = 𝐶))
3029cbvrexv 2512 . 2 (u B (𝐹u) = 𝐶x B (𝐹x) = 𝐶)
3127, 30syl6bb 185 1 ((𝐹 Fn A BA) → (𝐶 (𝐹B) ↔ x B (𝐹x) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  {cab 2008  wrex 2285  Vcvv 2535  wss 2894  dom cdm 4272  cima 4275  Fun wfun 4823   Fn wfn 4824  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  ssimaex  5159  rexima  5319  ralima  5320  f1elima  5337  ovelimab  5574
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