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Theorem preimafvelsetpreimafv 45110
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 6811 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
32sneqd 4582 . . . . . . 7 (𝑥 = 𝑋 → {(𝐹𝑥)} = {(𝐹𝑋)})
43imaeq2d 5986 . . . . . 6 (𝑥 = 𝑋 → (𝐹 “ {(𝐹𝑥)}) = (𝐹 “ {(𝐹𝑋)}))
54eqeq2d 2747 . . . . 5 (𝑥 = 𝑋 → ((𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}) ↔ (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑋)})))
65adantl 482 . . . 4 ((𝑋𝐴𝑥 = 𝑋) → ((𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}) ↔ (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑋)})))
7 eqidd 2737 . . . 4 (𝑋𝐴 → (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑋)}))
81, 6, 7rspcedvd 3571 . . 3 (𝑋𝐴 → ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}))
983ad2ant3 1134 . 2 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}))
10 fnex 7132 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
11 cnvexg 7817 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
12 imaexg 7808 . . . . 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 45106 . . 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 256 1 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  {cab 2713  wrex 3070  Vcvv 3440  {csn 4570  ccnv 5606  cima 5610   Fn wfn 6460  cfv 6465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473
This theorem is referenced by:  imasetpreimafvbijlemfo  45127  fundcmpsurbijinjpreimafv  45129
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