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Theorem elsetpreimafveqfv 45674
Description: The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
elsetpreimafveqfv ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑋𝑆𝑌𝑆)) → (𝐹𝑋) = (𝐹𝑌))
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 elsetpreimafveqfv
StepHypRef Expression
1 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
21elsetpreimafvbi 45673 . . . 4 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
3 simpr 486 . . . . 5 ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)) → (𝐹𝑌) = (𝐹𝑋))
43eqcomd 2739 . . . 4 ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)) → (𝐹𝑋) = (𝐹𝑌))
52, 4syl6bi 253 . . 3 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 → (𝐹𝑋) = (𝐹𝑌)))
653exp 1120 . 2 (𝐹 Fn 𝐴 → (𝑆𝑃 → (𝑋𝑆 → (𝑌𝑆 → (𝐹𝑋) = (𝐹𝑌)))))
763imp2 1350 1 ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑋𝑆𝑌𝑆)) → (𝐹𝑋) = (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wrex 3070  {csn 4590  ccnv 5636  cima 5640   Fn wfn 6495  cfv 6500
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 5260  ax-nul 5267  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508
This theorem is referenced by:  imasetpreimafvbijlemfv  45684
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