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Theorem uniimaprimaeqfv 45820
Description: The union of the image of the preimage of a function value is the function value. (Contributed by AV, 12-Mar-2024.)
Assertion
Ref Expression
uniimaprimaeqfv ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑋)})) = (𝐹𝑋))

Proof of Theorem uniimaprimaeqfv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dffn3 6717 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
21biimpi 215 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
32adantr 481 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → 𝐹:𝐴⟶ran 𝐹)
4 cnvimass 6069 . . . . 5 (𝐹 “ {(𝐹𝑋)}) ⊆ dom 𝐹
5 fndm 6641 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
64, 5sseqtrid 4030 . . . 4 (𝐹 Fn 𝐴 → (𝐹 “ {(𝐹𝑋)}) ⊆ 𝐴)
76adantr 481 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ⊆ 𝐴)
8 preimafvsnel 45817 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → 𝑋 ∈ (𝐹 “ {(𝐹𝑋)}))
93, 7, 83jca 1128 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹:𝐴⟶ran 𝐹 ∧ (𝐹 “ {(𝐹𝑋)}) ⊆ 𝐴𝑋 ∈ (𝐹 “ {(𝐹𝑋)})))
10 fniniseg 7046 . . . . 5 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹 “ {(𝐹𝑋)}) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐹𝑋))))
1110adantr 481 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑥 ∈ (𝐹 “ {(𝐹𝑋)}) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐹𝑋))))
12 simpr 485 . . . 4 ((𝑥𝐴 ∧ (𝐹𝑥) = (𝐹𝑋)) → (𝐹𝑥) = (𝐹𝑋))
1311, 12syl6bi 252 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑥 ∈ (𝐹 “ {(𝐹𝑋)}) → (𝐹𝑥) = (𝐹𝑋)))
1413ralrimiv 3144 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → ∀𝑥 ∈ (𝐹 “ {(𝐹𝑋)})(𝐹𝑥) = (𝐹𝑋))
15 uniimafveqt 45819 . 2 ((𝐹:𝐴⟶ran 𝐹 ∧ (𝐹 “ {(𝐹𝑋)}) ⊆ 𝐴𝑋 ∈ (𝐹 “ {(𝐹𝑋)})) → (∀𝑥 ∈ (𝐹 “ {(𝐹𝑋)})(𝐹𝑥) = (𝐹𝑋) → (𝐹 “ (𝐹 “ {(𝐹𝑋)})) = (𝐹𝑋)))
169, 14, 15sylc 65 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑋)})) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  wss 3944  {csn 4622   cuni 4901  ccnv 5668  dom cdm 5669  ran crn 5670  cima 5672   Fn wfn 6527  wf 6528  cfv 6532
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540
This theorem is referenced by:  imasetpreimafvbijlemfo  45843  fundcmpsurbijinjpreimafv  45845
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