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Theorem preimafvelsetpreimafv 47375
Description: The preimage of a function value is an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
preimafvelsetpreimafv ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ∈ 𝑃)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑋,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem preimafvelsetpreimafv
StepHypRef Expression
1 id 22 . . . 4 (𝑋𝐴𝑋𝐴)
2 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
32sneqd 4638 . . . . . . 7 (𝑥 = 𝑋 → {(𝐹𝑥)} = {(𝐹𝑋)})
43imaeq2d 6078 . . . . . 6 (𝑥 = 𝑋 → (𝐹 “ {(𝐹𝑥)}) = (𝐹 “ {(𝐹𝑋)}))
54eqeq2d 2748 . . . . 5 (𝑥 = 𝑋 → ((𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}) ↔ (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑋)})))
65adantl 481 . . . 4 ((𝑋𝐴𝑥 = 𝑋) → ((𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}) ↔ (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑋)})))
7 eqidd 2738 . . . 4 (𝑋𝐴 → (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑋)}))
81, 6, 7rspcedvd 3624 . . 3 (𝑋𝐴 → ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}))
983ad2ant3 1136 . 2 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}))
10 fnex 7237 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
11 cnvexg 7946 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
12 imaexg 7935 . . . . 5 (𝐹 ∈ V → (𝐹 “ {(𝐹𝑋)}) ∈ V)
1310, 11, 123syl 18 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹 “ {(𝐹𝑋)}) ∈ V)
14133adant3 1133 . . 3 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ∈ V)
15 setpreimafvex.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1615elsetpreimafvb 47371 . . 3 ((𝐹 “ {(𝐹𝑋)}) ∈ V → ((𝐹 “ {(𝐹𝑋)}) ∈ 𝑃 ↔ ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)})))
1714, 16syl 17 . 2 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → ((𝐹 “ {(𝐹𝑋)}) ∈ 𝑃 ↔ ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)})))
189, 17mpbird 257 1 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  {csn 4626  ccnv 5684  cima 5688   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  imasetpreimafvbijlemfo  47392  fundcmpsurbijinjpreimafv  47394
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