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Theorem preimafvelsetpreimafv 47313
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 6907 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
32sneqd 4643 . . . . . . 7 (𝑥 = 𝑋 → {(𝐹𝑥)} = {(𝐹𝑋)})
43imaeq2d 6080 . . . . . 6 (𝑥 = 𝑋 → (𝐹 “ {(𝐹𝑥)}) = (𝐹 “ {(𝐹𝑋)}))
54eqeq2d 2746 . . . . 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 2736 . . . 4 (𝑋𝐴 → (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑋)}))
81, 6, 7rspcedvd 3624 . . 3 (𝑋𝐴 → ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}))
983ad2ant3 1134 . 2 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}))
10 fnex 7237 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
11 cnvexg 7947 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
12 imaexg 7936 . . . . 5 (𝐹 ∈ V → (𝐹 “ {(𝐹𝑋)}) ∈ V)
1310, 11, 123syl 18 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹 “ {(𝐹𝑋)}) ∈ V)
14133adant3 1131 . . 3 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ∈ V)
15 setpreimafvex.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1615elsetpreimafvb 47309 . . 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 1086   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  {csn 4631  ccnv 5688  cima 5692   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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
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-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  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-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  imasetpreimafvbijlemfo  47330  fundcmpsurbijinjpreimafv  47332
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