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Theorem preimafvelsetpreimafv 45570
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 6842 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
32sneqd 4598 . . . . . . 7 (𝑥 = 𝑋 → {(𝐹𝑥)} = {(𝐹𝑋)})
43imaeq2d 6013 . . . . . 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 3583 . . 3 (𝑋𝐴 → ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}))
983ad2ant3 1135 . 2 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → ∃𝑥𝐴 (𝐹 “ {(𝐹𝑋)}) = (𝐹 “ {(𝐹𝑥)}))
10 fnex 7167 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
11 cnvexg 7861 . . . . 5 (𝐹 ∈ V → 𝐹 ∈ V)
12 imaexg 7852 . . . . 5 (𝐹 ∈ V → (𝐹 “ {(𝐹𝑋)}) ∈ V)
1310, 11, 123syl 18 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹 “ {(𝐹𝑋)}) ∈ V)
14133adant3 1132 . . 3 ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ∈ V)
15 setpreimafvex.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1615elsetpreimafvb 45566 . . 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 1087   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  Vcvv 3445  {csn 4586  ccnv 5632  cima 5636   Fn wfn 6491  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504
This theorem is referenced by:  imasetpreimafvbijlemfo  45587  fundcmpsurbijinjpreimafv  45589
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