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Theorem fvelima2 41885
Description: Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Assertion
Ref Expression
fvelima2 ((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹

Proof of Theorem fvelima2
StepHypRef Expression
1 elimag 5904 . . . 4 (𝐵 ∈ (𝐹𝐶) → (𝐵 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 𝑥𝐹𝐵))
21ibi 270 . . 3 (𝐵 ∈ (𝐹𝐶) → ∃𝑥𝐶 𝑥𝐹𝐵)
3 df-rex 3115 . . 3 (∃𝑥𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
42, 3sylib 221 . 2 (𝐵 ∈ (𝐹𝐶) → ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
5 fnbr 6434 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → 𝑥𝐴)
65adantrl 715 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐴)
7 simprl 770 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐶)
86, 7elind 4124 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴𝐶))
9 fnfun 6427 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
10 funbrfv 6695 . . . . . . . . . 10 (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹𝑥) = 𝐵))
1110imp 410 . . . . . . . . 9 ((Fun 𝐹𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
129, 11sylan 583 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
1312adantrl 715 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝐹𝑥) = 𝐵)
148, 13jca 515 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1514ex 416 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐶𝑥𝐹𝐵) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1615eximdv 1918 . . . 4 (𝐹 Fn 𝐴 → (∃𝑥(𝑥𝐶𝑥𝐹𝐵) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1716imp 410 . . 3 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
18 df-rex 3115 . . 3 (∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1917, 18sylibr 237 . 2 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
204, 19sylan2 595 1 ((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  wrex 3110  cin 3883   class class class wbr 5033  cima 5526  Fun wfun 6322   Fn wfn 6323  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  limsupresxr  42395  liminfresxr  42396  liminfvalxr  42412
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