Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvelimad Structured version   Visualization version   GIF version

 Description: Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
Assertion
Ref Expression
fvelimad (𝜑 → ∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvelimad.c . . . 4 (𝜑𝐶 ∈ (𝐹𝐵))
2 elimag 5505 . . . . 5 (𝐶 ∈ (𝐹𝐵) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑦𝐹𝐶))
32ibi 256 . . . 4 (𝐶 ∈ (𝐹𝐵) → ∃𝑦𝐵 𝑦𝐹𝐶)
41, 3syl 17 . . 3 (𝜑 → ∃𝑦𝐵 𝑦𝐹𝐶)
5 nfv 1883 . . . 4 𝑦𝜑
6 nfre1 3034 . . . 4 𝑦𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶
7 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
87a1i 11 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ V)
91adantr 480 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝐶 ∈ (𝐹𝐵))
10 simpr 476 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦𝐹𝐶)
11 breldmg 5362 . . . . . . . . . 10 ((𝑦 ∈ V ∧ 𝐶 ∈ (𝐹𝐵) ∧ 𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
128, 9, 10, 11syl3anc 1366 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
13 fvelimad.f . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
1413fndmd 39755 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
1514adantr 480 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → dom 𝐹 = 𝐴)
1612, 15eleqtrd 2732 . . . . . . . 8 ((𝜑𝑦𝐹𝐶) → 𝑦𝐴)
17163adant2 1100 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐴)
18 simp2 1082 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐵)
1917, 18elind 3831 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦 ∈ (𝐴𝐵))
20 fnfun 6026 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
2113, 20syl 17 . . . . . . . 8 (𝜑 → Fun 𝐹)
22213ad2ant1 1102 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → Fun 𝐹)
23 simp3 1083 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐹𝐶)
24 funbrfv 6272 . . . . . . 7 (Fun 𝐹 → (𝑦𝐹𝐶 → (𝐹𝑦) = 𝐶))
2522, 23, 24sylc 65 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → (𝐹𝑦) = 𝐶)
26 rspe 3032 . . . . . 6 ((𝑦 ∈ (𝐴𝐵) ∧ (𝐹𝑦) = 𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
2719, 25, 26syl2anc 694 . . . . 5 ((𝜑𝑦𝐵𝑦𝐹𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
28273exp 1283 . . . 4 (𝜑 → (𝑦𝐵 → (𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)))
295, 6, 28rexlimd 3055 . . 3 (𝜑 → (∃𝑦𝐵 𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶))
304, 29mpd 15 . 2 (𝜑 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
31 nfv 1883 . . 3 𝑦(𝐹𝑥) = 𝐶
32 fvelimad.x . . . . 5 𝑥𝐹
33 nfcv 2793 . . . . 5 𝑥𝑦
3432, 33nffv 6236 . . . 4 𝑥(𝐹𝑦)
35 nfcv 2793 . . . 4 𝑥𝐶
3634, 35nfeq 2805 . . 3 𝑥(𝐹𝑦) = 𝐶
37 fveq2 6229 . . . 4 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
3837eqeq1d 2653 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝑦) = 𝐶))
3931, 36, 38cbvrex 3198 . 2 (∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶 ↔ ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
4030, 39sylibr 224 1 (𝜑 → ∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Ⅎwnfc 2780  ∃wrex 2942  Vcvv 3231   ∩ cin 3606   class class class wbr 4685  dom cdm 5143   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934 This theorem is referenced by:  limsupmnflem  40270  liminfvalxr  40333
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