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Theorem elsetpreimafvbi 44377
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 6837 . . . . . 6 (𝐹 Fn 𝐴 → (𝑋 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑋𝐴 ∧ (𝐹𝑋) = (𝐹𝑥))))
2 fniniseg 6837 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥))))
3 eqeq2 2750 . . . . . . . . . . 11 ((𝐹𝑥) = (𝐹𝑋) → ((𝐹𝑌) = (𝐹𝑥) ↔ (𝐹𝑌) = (𝐹𝑋)))
43anbi2d 632 . . . . . . . . . 10 ((𝐹𝑥) = (𝐹𝑋) → ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥)) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
54eqcoms 2746 . . . . . . . . 9 ((𝐹𝑋) = (𝐹𝑥) → ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑥)) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
62, 5sylan9bb 513 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝐹𝑋) = (𝐹𝑥)) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
76ex 416 . . . . . . 7 (𝐹 Fn 𝐴 → ((𝐹𝑋) = (𝐹𝑥) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
87adantld 494 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑋𝐴 ∧ (𝐹𝑋) = (𝐹𝑥)) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
91, 8sylbid 243 . . . . 5 (𝐹 Fn 𝐴 → (𝑋 ∈ (𝐹 “ {(𝐹𝑥)}) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
10 eleq2 2821 . . . . . 6 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝑋𝑆𝑋 ∈ (𝐹 “ {(𝐹𝑥)})))
11 eleq2 2821 . . . . . . 7 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝑌𝑆𝑌 ∈ (𝐹 “ {(𝐹𝑥)})))
1211bibi1d 347 . . . . . 6 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → ((𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))) ↔ (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))))
1310, 12imbi12d 348 . . . . 5 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → ((𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)))) ↔ (𝑋 ∈ (𝐹 “ {(𝐹𝑥)}) → (𝑌 ∈ (𝐹 “ {(𝐹𝑥)}) ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
149, 13syl5ibr 249 . . . 4 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴 → (𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
1514rexlimivw 3192 . . 3 (∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴 → (𝑋𝑆 → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))))
16 setpreimafvex.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1716elsetpreimafv 44371 . . 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  {cab 2716  wrex 3054  {csn 4516  ccnv 5524  cima 5528   Fn wfn 6334  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-fv 6347
This theorem is referenced by:  elsetpreimafveqfv  44378  eqfvelsetpreimafv  44379  elsetpreimafvrab  44380  imaelsetpreimafv  44381
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