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Theorem uniimaprimaeqfv 46535
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 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ βˆͺ (𝐹 β€œ (◑𝐹 β€œ {(πΉβ€˜π‘‹)})) = (πΉβ€˜π‘‹))

Proof of Theorem uniimaprimaeqfv
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 dffn3 6720 . . . . 5 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
21biimpi 215 . . . 4 (𝐹 Fn 𝐴 β†’ 𝐹:𝐴⟢ran 𝐹)
32adantr 480 . . 3 ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ 𝐹:𝐴⟢ran 𝐹)
4 cnvimass 6070 . . . . 5 (◑𝐹 β€œ {(πΉβ€˜π‘‹)}) βŠ† dom 𝐹
5 fndm 6642 . . . . 5 (𝐹 Fn 𝐴 β†’ dom 𝐹 = 𝐴)
64, 5sseqtrid 4026 . . . 4 (𝐹 Fn 𝐴 β†’ (◑𝐹 β€œ {(πΉβ€˜π‘‹)}) βŠ† 𝐴)
76adantr 480 . . 3 ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ (◑𝐹 β€œ {(πΉβ€˜π‘‹)}) βŠ† 𝐴)
8 preimafvsnel 46532 . . 3 ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ 𝑋 ∈ (◑𝐹 β€œ {(πΉβ€˜π‘‹)}))
93, 7, 83jca 1125 . 2 ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ (𝐹:𝐴⟢ran 𝐹 ∧ (◑𝐹 β€œ {(πΉβ€˜π‘‹)}) βŠ† 𝐴 ∧ 𝑋 ∈ (◑𝐹 β€œ {(πΉβ€˜π‘‹)})))
10 fniniseg 7051 . . . . 5 (𝐹 Fn 𝐴 β†’ (π‘₯ ∈ (◑𝐹 β€œ {(πΉβ€˜π‘‹)}) ↔ (π‘₯ ∈ 𝐴 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))))
1110adantr 480 . . . 4 ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ (π‘₯ ∈ (◑𝐹 β€œ {(πΉβ€˜π‘‹)}) ↔ (π‘₯ ∈ 𝐴 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))))
12 simpr 484 . . . 4 ((π‘₯ ∈ 𝐴 ∧ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹)) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
1311, 12syl6bi 253 . . 3 ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ (π‘₯ ∈ (◑𝐹 β€œ {(πΉβ€˜π‘‹)}) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹)))
1413ralrimiv 3137 . 2 ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ βˆ€π‘₯ ∈ (◑𝐹 β€œ {(πΉβ€˜π‘‹)})(πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
15 uniimafveqt 46534 . 2 ((𝐹:𝐴⟢ran 𝐹 ∧ (◑𝐹 β€œ {(πΉβ€˜π‘‹)}) βŠ† 𝐴 ∧ 𝑋 ∈ (◑𝐹 β€œ {(πΉβ€˜π‘‹)})) β†’ (βˆ€π‘₯ ∈ (◑𝐹 β€œ {(πΉβ€˜π‘‹)})(πΉβ€˜π‘₯) = (πΉβ€˜π‘‹) β†’ βˆͺ (𝐹 β€œ (◑𝐹 β€œ {(πΉβ€˜π‘‹)})) = (πΉβ€˜π‘‹)))
169, 14, 15sylc 65 1 ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) β†’ βˆͺ (𝐹 β€œ (◑𝐹 β€œ {(πΉβ€˜π‘‹)})) = (πΉβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3940  {csn 4620  βˆͺ cuni 4899  β—‘ccnv 5665  dom cdm 5666  ran crn 5667   β€œ cima 5669   Fn wfn 6528  βŸΆwf 6529  β€˜cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541
This theorem is referenced by:  imasetpreimafvbijlemfo  46558  fundcmpsurbijinjpreimafv  46560
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