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Theorem elsetpreimafveq 44849
Description: If two preimages of function values contain elements with identical function values, then both preimages are equal. (Contributed by AV, 8-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
elsetpreimafveq ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑆 = 𝑅))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑅,𝑧   𝑥,𝑆,𝑧   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem elsetpreimafveq
StepHypRef Expression
1 eqeq2 2750 . . . . 5 ((𝐹𝑋) = (𝐹𝑌) → ((𝐹𝑥) = (𝐹𝑋) ↔ (𝐹𝑥) = (𝐹𝑌)))
21rabbidv 3414 . . . 4 ((𝐹𝑋) = (𝐹𝑌) → {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)} = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑌)})
32adantl 482 . . 3 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)} = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑌)})
4 id 22 . . . . . 6 (𝐹 Fn 𝐴𝐹 Fn 𝐴)
5 simpl 483 . . . . . 6 ((𝑆𝑃𝑅𝑃) → 𝑆𝑃)
6 simpl 483 . . . . . 6 ((𝑋𝑆𝑌𝑅) → 𝑋𝑆)
74, 5, 63anim123i 1150 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → (𝐹 Fn 𝐴𝑆𝑃𝑋𝑆))
87adantr 481 . . . 4 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → (𝐹 Fn 𝐴𝑆𝑃𝑋𝑆))
9 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
109elsetpreimafvrab 44846 . . . 4 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → 𝑆 = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)})
118, 10syl 17 . . 3 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → 𝑆 = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)})
12 simpr 485 . . . . . 6 ((𝑆𝑃𝑅𝑃) → 𝑅𝑃)
13 simpr 485 . . . . . 6 ((𝑋𝑆𝑌𝑅) → 𝑌𝑅)
144, 12, 133anim123i 1150 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → (𝐹 Fn 𝐴𝑅𝑃𝑌𝑅))
1514adantr 481 . . . 4 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → (𝐹 Fn 𝐴𝑅𝑃𝑌𝑅))
169elsetpreimafvrab 44846 . . . 4 ((𝐹 Fn 𝐴𝑅𝑃𝑌𝑅) → 𝑅 = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑌)})
1715, 16syl 17 . . 3 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → 𝑅 = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑌)})
183, 11, 173eqtr4d 2788 . 2 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → 𝑆 = 𝑅)
1918ex 413 1 ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑆 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  {cab 2715  wrex 3065  {crab 3068  {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:  imasetpreimafvbijlemf1  44856
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