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Theorem elsetpreimafvbi 45994
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 7057 . . . . . 6 (𝐹 Fn 𝐴 → (𝑋 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑋𝐴 ∧ (𝐹𝑋) = (𝐹𝑥))))
2 fniniseg 7057 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥))))
3 eqeq2 2745 . . . . . . . . . . 11 ((𝐹𝑥) = (𝐹𝑋) → ((𝐹𝑌) = (𝐹𝑥) ↔ (𝐹𝑌) = (𝐹𝑋)))
43anbi2d 630 . . . . . . . . . 10 ((𝐹𝑥) = (𝐹𝑋) → ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥)) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
54eqcoms 2741 . . . . . . . . 9 ((𝐹𝑋) = (𝐹𝑥) → ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥)) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
62, 5sylan9bb 511 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝐹𝑋) = (𝐹𝑥)) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
76ex 414 . . . . . . 7 (𝐹 Fn 𝐴 → ((𝐹𝑋) = (𝐹𝑥) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
87adantld 492 . . . . . 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 2823 . . . . . 6 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝑋𝑆𝑋 ∈ (𝐹 “ {(𝐹𝑥)})))
11 eleq2 2823 . . . . . . 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, 13imbitrrid 245 . . . 4 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴 → (𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
1514rexlimivw 3152 . . 3 (∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴 → (𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
16 setpreimafvex.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1716elsetpreimafv 45988 . . 3 (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))
1815, 17syl11 33 . 2 (𝐹 Fn 𝐴 → (𝑆𝑃 → (𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
19183imp 1112 1 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  {csn 4627  ccnv 5674  cima 5678   Fn wfn 6535  cfv 6540
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-fv 6548
This theorem is referenced by:  elsetpreimafveqfv  45995  eqfvelsetpreimafv  45996  elsetpreimafvrab  45997  imaelsetpreimafv  45998
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