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Theorem elsetpreimafvbi 44843
Description: An element of the preimage of a function value is an element of the domain of the function with the same value as another element of the preimage. (Contributed by AV, 9-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
elsetpreimafvbi ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem elsetpreimafvbi
StepHypRef Expression
1 fniniseg 6937 . . . . . 6 (𝐹 Fn 𝐴 → (𝑋 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑋𝐴 ∧ (𝐹𝑋) = (𝐹𝑥))))
2 fniniseg 6937 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥))))
3 eqeq2 2750 . . . . . . . . . . 11 ((𝐹𝑥) = (𝐹𝑋) → ((𝐹𝑌) = (𝐹𝑥) ↔ (𝐹𝑌) = (𝐹𝑋)))
43anbi2d 629 . . . . . . . . . 10 ((𝐹𝑥) = (𝐹𝑋) → ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥)) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
54eqcoms 2746 . . . . . . . . 9 ((𝐹𝑋) = (𝐹𝑥) → ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥)) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
62, 5sylan9bb 510 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝐹𝑋) = (𝐹𝑥)) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
76ex 413 . . . . . . 7 (𝐹 Fn 𝐴 → ((𝐹𝑋) = (𝐹𝑥) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
87adantld 491 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑋𝐴 ∧ (𝐹𝑋) = (𝐹𝑥)) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
91, 8sylbid 239 . . . . 5 (𝐹 Fn 𝐴 → (𝑋 ∈ (𝐹 “ {(𝐹𝑥)}) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
10 eleq2 2827 . . . . . 6 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝑋𝑆𝑋 ∈ (𝐹 “ {(𝐹𝑥)})))
11 eleq2 2827 . . . . . . 7 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝑌𝑆𝑌 ∈ (𝐹 “ {(𝐹𝑥)})))
1211bibi1d 344 . . . . . 6 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → ((𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))) ↔ (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
1310, 12imbi12d 345 . . . . 5 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → ((𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))) ↔ (𝑋 ∈ (𝐹 “ {(𝐹𝑥)}) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
149, 13syl5ibr 245 . . . 4 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴 → (𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
1514rexlimivw 3211 . . 3 (∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴 → (𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
16 setpreimafvex.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1716elsetpreimafv 44837 . . 3 (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))
1815, 17syl11 33 . 2 (𝐹 Fn 𝐴 → (𝑆𝑃 → (𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
19183imp 1110 1 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  {cab 2715  wrex 3065  {csn 4561  ccnv 5588  cima 5592   Fn wfn 6428  cfv 6433
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441
This theorem is referenced by:  elsetpreimafveqfv  44844  eqfvelsetpreimafv  44845  elsetpreimafvrab  44846  imaelsetpreimafv  44847
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