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Theorem fvelimad 6960
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 6064 . . . . 5 (𝐶 ∈ (𝐹𝐵) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑦𝐹𝐶))
32ibi 267 . . . 4 (𝐶 ∈ (𝐹𝐵) → ∃𝑦𝐵 𝑦𝐹𝐶)
41, 3syl 17 . . 3 (𝜑 → ∃𝑦𝐵 𝑦𝐹𝐶)
5 nfv 1918 . . . 4 𝑦𝜑
6 nfre1 3283 . . . 4 𝑦𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶
7 vex 3479 . . . . . . . . . . 11 𝑦 ∈ V
87a1i 11 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ V)
91adantr 482 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝐶 ∈ (𝐹𝐵))
10 simpr 486 . . . . . . . . . 10 ((𝜑𝑦𝐹𝐶) → 𝑦𝐹𝐶)
118, 9, 10breldmd 5913 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → 𝑦 ∈ dom 𝐹)
12 fvelimad.f . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
1312fndmd 6655 . . . . . . . . . 10 (𝜑 → dom 𝐹 = 𝐴)
1413adantr 482 . . . . . . . . 9 ((𝜑𝑦𝐹𝐶) → dom 𝐹 = 𝐴)
1511, 14eleqtrd 2836 . . . . . . . 8 ((𝜑𝑦𝐹𝐶) → 𝑦𝐴)
16153adant2 1132 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐴)
17 simp2 1138 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐵)
1816, 17elind 4195 . . . . . 6 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦 ∈ (𝐴𝐵))
19 fnfun 6650 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
2012, 19syl 17 . . . . . . . 8 (𝜑 → Fun 𝐹)
21203ad2ant1 1134 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → Fun 𝐹)
22 simp3 1139 . . . . . . 7 ((𝜑𝑦𝐵𝑦𝐹𝐶) → 𝑦𝐹𝐶)
23 funbrfv 6943 . . . . . . 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 585 . . . . 5 ((𝜑𝑦𝐵𝑦𝐹𝐶) → ∃𝑦 ∈ (𝐴𝐵)(𝐹𝑦) = 𝐶)
27263exp 1120 . . . 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 1918 . . 3 𝑦(𝐹𝑥) = 𝐶
31 fvelimad.x . . . . 5 𝑥𝐹
32 nfcv 2904 . . . . 5 𝑥𝑦
3331, 32nffv 6902 . . . 4 𝑥(𝐹𝑦)
3433nfeq1 2919 . . 3 𝑥(𝐹𝑦) = 𝐶
35 fveqeq2 6901 . . 3 (𝑥 = 𝑦 → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝑦) = 𝐶))
3630, 34, 35cbvrexw 3305 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wnfc 2884  wrex 3071  Vcvv 3475  cin 3948   class class class wbr 5149  dom cdm 5677  cima 5680  Fun wfun 6538   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552
This theorem is referenced by:  cyc3evpm  32309  cycpmgcl  32312  cycpmconjslem2  32314  cyc3conja  32316  limsupmnflem  44436  liminfvalxr  44499
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