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Theorem elsetpreimafveq 44431
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 2751 . . . . 5 ((𝐹𝑋) = (𝐹𝑌) → ((𝐹𝑥) = (𝐹𝑋) ↔ (𝐹𝑥) = (𝐹𝑌)))
21rabbidv 3382 . . . 4 ((𝐹𝑋) = (𝐹𝑌) → {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)} = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑌)})
32adantl 485 . . 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 486 . . . . . 6 ((𝑆𝑃𝑅𝑃) → 𝑆𝑃)
6 simpl 486 . . . . . 6 ((𝑋𝑆𝑌𝑅) → 𝑋𝑆)
74, 5, 63anim123i 1152 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → (𝐹 Fn 𝐴𝑆𝑃𝑋𝑆))
87adantr 484 . . . 4 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → (𝐹 Fn 𝐴𝑆𝑃𝑋𝑆))
9 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
109elsetpreimafvrab 44428 . . . 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 488 . . . . . 6 ((𝑆𝑃𝑅𝑃) → 𝑅𝑃)
13 simpr 488 . . . . . 6 ((𝑋𝑆𝑌𝑅) → 𝑌𝑅)
144, 12, 133anim123i 1152 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → (𝐹 Fn 𝐴𝑅𝑃𝑌𝑅))
1514adantr 484 . . . 4 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → (𝐹 Fn 𝐴𝑅𝑃𝑌𝑅))
169elsetpreimafvrab 44428 . . . 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 2784 . 2 (((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) ∧ (𝐹𝑋) = (𝐹𝑌)) → 𝑆 = 𝑅)
1918ex 416 1 ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑆 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  {cab 2717  wrex 3055  {crab 3058  {csn 4526  ccnv 5534  cima 5538   Fn wfn 6345  cfv 6350
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 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
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 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-fv 6358
This theorem is referenced by:  imasetpreimafvbijlemf1  44438
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