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Theorem uniimaprimaeqfv 44295
 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 6514 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
21biimpi 219 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
32adantr 484 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → 𝐹:𝐴⟶ran 𝐹)
4 cnvimass 5925 . . . . 5 (𝐹 “ {(𝐹𝑋)}) ⊆ dom 𝐹
5 fndm 6440 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
64, 5sseqtrid 3946 . . . 4 (𝐹 Fn 𝐴 → (𝐹 “ {(𝐹𝑋)}) ⊆ 𝐴)
76adantr 484 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ⊆ 𝐴)
8 preimafvsnel 44292 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → 𝑋 ∈ (𝐹 “ {(𝐹𝑋)}))
93, 7, 83jca 1125 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → (𝐹:𝐴⟶ran 𝐹 ∧ (𝐹 “ {(𝐹𝑋)}) ⊆ 𝐴𝑋 ∈ (𝐹 “ {(𝐹𝑋)})))
10 fniniseg 6825 . . . . 5 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹 “ {(𝐹𝑋)}) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐹𝑋))))
1110adantr 484 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑥 ∈ (𝐹 “ {(𝐹𝑋)}) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = (𝐹𝑋))))
12 simpr 488 . . . 4 ((𝑥𝐴 ∧ (𝐹𝑥) = (𝐹𝑋)) → (𝐹𝑥) = (𝐹𝑋))
1311, 12syl6bi 256 . . 3 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑥 ∈ (𝐹 “ {(𝐹𝑋)}) → (𝐹𝑥) = (𝐹𝑋)))
1413ralrimiv 3112 . 2 ((𝐹 Fn 𝐴𝑋𝐴) → ∀𝑥 ∈ (𝐹 “ {(𝐹𝑋)})(𝐹𝑥) = (𝐹𝑋))
15 uniimafveqt 44294 . 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3860  {csn 4525  ∪ cuni 4801  ◡ccnv 5526  dom cdm 5527  ran crn 5528   “ cima 5530   Fn wfn 6334  ⟶wf 6335  ‘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-ne 2952  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-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  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-f 6343  df-fv 6347 This theorem is referenced by:  imasetpreimafvbijlemfo  44318  fundcmpsurbijinjpreimafv  44320
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