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Theorem fvelimad 6976
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 6084 . . . . 5 (𝐶 ∈ (𝐹𝐵) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑦𝐹𝐶))
32ibi 267 . . . 4 (𝐶 ∈ (𝐹𝐵) → ∃𝑦𝐵 𝑦𝐹𝐶)
41, 3syl 17 . . 3 (𝜑 → ∃𝑦𝐵 𝑦𝐹𝐶)
5 nfv 1912 . . . 4 𝑦𝜑
6 nfre1 3283 . . . 4 𝑦𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶
7 vex 3482 . . . . . . . . . . 11 𝑦 ∈ V
87a1i 11 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ V)
91adantr 480 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝐶 ∈ (𝐹𝐵))
10 simpr 484 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦𝐹𝐶)
118, 9, 10breldmd 5926 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
12 fvelimad.f . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
1312fndmd 6674 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
1413adantr 480 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → dom 𝐹 = 𝐴)
1511, 14eleqtrd 2841 . . . . . . . 8 ((𝜑𝑦𝐹𝐶) → 𝑦𝐴)
16153adant2 1130 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐴)
17 simp2 1136 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐵)
1816, 17elind 4210 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦 ∈ (𝐴𝐵))
19 fnfun 6669 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
2012, 19syl 17 . . . . . . . 8 (𝜑 → Fun 𝐹)
21203ad2ant1 1132 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → Fun 𝐹)
22 simp3 1137 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐹𝐶)
23 funbrfv 6958 . . . . . . 7 (Fun 𝐹 → (𝑦𝐹𝐶 → (𝐹𝑦) = 𝐶))
2421, 22, 23sylc 65 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → (𝐹𝑦) = 𝐶)
25 rspe 3247 . . . . . 6 ((𝑦 ∈ (𝐴𝐵) ∧ (𝐹𝑦) = 𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
2618, 24, 25syl2anc 584 . . . . 5 ((𝜑𝑦𝐵𝑦𝐹𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
27263exp 1118 . . . 4 (𝜑 → (𝑦𝐵 → (𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)))
285, 6, 27rexlimd 3264 . . 3 (𝜑 → (∃𝑦𝐵 𝑦𝐹𝐶 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶))
294, 28mpd 15 . 2 (𝜑 → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
30 nfv 1912 . . 3 𝑦(𝐹𝑥) = 𝐶
31 fvelimad.x . . . . 5 𝑥𝐹
32 nfcv 2903 . . . . 5 𝑥𝑦
3331, 32nffv 6917 . . . 4 𝑥(𝐹𝑦)
3433nfeq1 2919 . . 3 𝑥(𝐹𝑦) = 𝐶
35 fveqeq2 6916 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝑦) = 𝐶))
3630, 34, 35cbvrexw 3305 . 2 (∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶 ↔ ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
3729, 36sylibr 234 1 (𝜑 → ∃𝑥 ∈ (𝐴𝐵)(𝐹𝑥) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wnfc 2888  wrex 3068  Vcvv 3478  cin 3962   class class class wbr 5148  dom cdm 5689  cima 5692  Fun wfun 6557   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  cyc3evpm  33153  cycpmgcl  33156  cycpmconjslem2  33158  cyc3conja  33160  limsupmnflem  45676  liminfvalxr  45739
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