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Theorem fvelimad 6779
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 5933 . . . . 5 (𝐶 ∈ (𝐹𝐵) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑦𝐹𝐶))
32ibi 270 . . . 4 (𝐶 ∈ (𝐹𝐵) → ∃𝑦𝐵 𝑦𝐹𝐶)
41, 3syl 17 . . 3 (𝜑 → ∃𝑦𝐵 𝑦𝐹𝐶)
5 nfv 1922 . . . 4 𝑦𝜑
6 nfre1 3225 . . . 4 𝑦𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶
7 vex 3412 . . . . . . . . . . 11 𝑦 ∈ V
87a1i 11 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ V)
91adantr 484 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝐶 ∈ (𝐹𝐵))
10 simpr 488 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦𝐹𝐶)
118, 9, 10breldmd 5781 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
12 fvelimad.f . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
1312fndmd 6483 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
1413adantr 484 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → dom 𝐹 = 𝐴)
1511, 14eleqtrd 2840 . . . . . . . 8 ((𝜑𝑦𝐹𝐶) → 𝑦𝐴)
16153adant2 1133 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐴)
17 simp2 1139 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐵)
1816, 17elind 4108 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦 ∈ (𝐴𝐵))
19 fnfun 6479 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
2012, 19syl 17 . . . . . . . 8 (𝜑 → Fun 𝐹)
21203ad2ant1 1135 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → Fun 𝐹)
22 simp3 1140 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐹𝐶)
23 funbrfv 6763 . . . . . . 7 (Fun 𝐹 → (𝑦𝐹𝐶 → (𝐹𝑦) = 𝐶))
2421, 22, 23sylc 65 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → (𝐹𝑦) = 𝐶)
25 rspe 3223 . . . . . 6 ((𝑦 ∈ (𝐴𝐵) ∧ (𝐹𝑦) = 𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
2618, 24, 25syl2anc 587 . . . . 5 ((𝜑𝑦𝐵𝑦𝐹𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
27263exp 1121 . . . 4 (𝜑 → (𝑦𝐵 → (𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)))
285, 6, 27rexlimd 3236 . . 3 (𝜑 → (∃𝑦𝐵 𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶))
294, 28mpd 15 . 2 (𝜑 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
30 nfv 1922 . . 3 𝑦(𝐹𝑥) = 𝐶
31 fvelimad.x . . . . 5 𝑥𝐹
32 nfcv 2904 . . . . 5 𝑥𝑦
3331, 32nffv 6727 . . . 4 𝑥(𝐹𝑦)
3433nfeq1 2919 . . 3 𝑥(𝐹𝑦) = 𝐶
35 fveqeq2 6726 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝑦) = 𝐶))
3630, 34, 35cbvrexw 3350 . 2 (∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶 ↔ ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
3729, 36sylibr 237 1 (𝜑 → ∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wnfc 2884  wrex 3062  Vcvv 3408  cin 3865   class class class wbr 5053  dom cdm 5551  cima 5554  Fun wfun 6374   Fn wfn 6375  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388
This theorem is referenced by:  cyc3evpm  31136  cycpmgcl  31139  cycpmconjslem2  31141  cyc3conja  31143  limsupmnflem  42936  liminfvalxr  42999
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