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Theorem fvelimad 6945
Description: Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fvelimad.x 𝑥𝐹
fvelimad.f (𝜑𝐹 Fn 𝐴)
fvelimad.c (𝜑𝐶 ∈ (𝐹𝐵))
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)

Proof of Theorem fvelimad
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvelimad.c . . . 4 (𝜑𝐶 ∈ (𝐹𝐵))
2 elimag 6053 . . . . 5 (𝐶 ∈ (𝐹𝐵) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑦𝐹𝐶))
32ibi 266 . . . 4 (𝐶 ∈ (𝐹𝐵) → ∃𝑦𝐵 𝑦𝐹𝐶)
41, 3syl 17 . . 3 (𝜑 → ∃𝑦𝐵 𝑦𝐹𝐶)
5 nfv 1917 . . . 4 𝑦𝜑
6 nfre1 3281 . . . 4 𝑦𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶
7 vex 3477 . . . . . . . . . . 11 𝑦 ∈ V
87a1i 11 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ V)
91adantr 481 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝐶 ∈ (𝐹𝐵))
10 simpr 485 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦𝐹𝐶)
118, 9, 10breldmd 5904 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
12 fvelimad.f . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
1312fndmd 6643 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
1413adantr 481 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → dom 𝐹 = 𝐴)
1511, 14eleqtrd 2834 . . . . . . . 8 ((𝜑𝑦𝐹𝐶) → 𝑦𝐴)
16153adant2 1131 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐴)
17 simp2 1137 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐵)
1816, 17elind 4190 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦 ∈ (𝐴𝐵))
19 fnfun 6638 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
2012, 19syl 17 . . . . . . . 8 (𝜑 → Fun 𝐹)
21203ad2ant1 1133 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → Fun 𝐹)
22 simp3 1138 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐹𝐶)
23 funbrfv 6929 . . . . . . 7 (Fun 𝐹 → (𝑦𝐹𝐶 → (𝐹𝑦) = 𝐶))
2421, 22, 23sylc 65 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → (𝐹𝑦) = 𝐶)
25 rspe 3245 . . . . . 6 ((𝑦 ∈ (𝐴𝐵) ∧ (𝐹𝑦) = 𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
2618, 24, 25syl2anc 584 . . . . 5 ((𝜑𝑦𝐵𝑦𝐹𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
27263exp 1119 . . . 4 (𝜑 → (𝑦𝐵 → (𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)))
285, 6, 27rexlimd 3262 . . 3 (𝜑 → (∃𝑦𝐵 𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶))
294, 28mpd 15 . 2 (𝜑 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
30 nfv 1917 . . 3 𝑦(𝐹𝑥) = 𝐶
31 fvelimad.x . . . . 5 𝑥𝐹
32 nfcv 2902 . . . . 5 𝑥𝑦
3331, 32nffv 6888 . . . 4 𝑥(𝐹𝑦)
3433nfeq1 2917 . . 3 𝑥(𝐹𝑦) = 𝐶
35 fveqeq2 6887 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝑦) = 𝐶))
3630, 34, 35cbvrexw 3303 . 2 (∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶 ↔ ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
3729, 36sylibr 233 1 (𝜑 → ∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wnfc 2882  wrex 3069  Vcvv 3473  cin 3943   class class class wbr 5141  dom cdm 5669  cima 5672  Fun wfun 6526   Fn wfn 6527  cfv 6532
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-fv 6540
This theorem is referenced by:  cyc3evpm  32180  cycpmgcl  32183  cycpmconjslem2  32185  cyc3conja  32187  limsupmnflem  44209  liminfvalxr  44272
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