Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elsetpreimafvrab Structured version   Visualization version   GIF version

Theorem elsetpreimafvrab 44307
 Description: An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
elsetpreimafvrab ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → 𝑆 = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)})
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧   𝑥,𝑋
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑋(𝑧)

Proof of Theorem elsetpreimafvrab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 setpreimafvex.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
21elsetpreimafvbi 44304 . . 3 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑦𝑆 ↔ (𝑦𝐴 ∧ (𝐹𝑦) = (𝐹𝑋))))
3 fveqeq2 6671 . . . 4 (𝑥 = 𝑦 → ((𝐹𝑥) = (𝐹𝑋) ↔ (𝐹𝑦) = (𝐹𝑋)))
43elrab 3604 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)} ↔ (𝑦𝐴 ∧ (𝐹𝑦) = (𝐹𝑋)))
52, 4bitr4di 292 . 2 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑦𝑆𝑦 ∈ {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)}))
65eqrdv 2756 1 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → 𝑆 = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071  {crab 3074  {csn 4525  ◡ccnv 5526   “ cima 5530   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 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-fv 6347 This theorem is referenced by:  elsetpreimafveq  44310
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