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Theorem uniimafveqt 44833
Description: The union of the image of a subset 𝑆 of the domain of a function with elements having the same function value is the function value at one of the elements of 𝑆. (Contributed by AV, 5-Mar-2024.)
Assertion
Ref Expression
uniimafveqt ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → (𝐹𝑆) = (𝐹𝑋)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem uniimafveqt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ffun 6603 . . . . . 6 (𝐹:𝐴𝐵 → Fun 𝐹)
213ad2ant1 1132 . . . . 5 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → Fun 𝐹)
32adantr 481 . . . 4 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → Fun 𝐹)
4 funiunfv 7121 . . . 4 (Fun 𝐹 𝑦𝑆 (𝐹𝑦) = (𝐹𝑆))
53, 4syl 17 . . 3 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑆))
6 simp3 1137 . . . 4 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → 𝑋𝑆)
7 fveqeq2 6783 . . . . . 6 (𝑥 = 𝑦 → ((𝐹𝑥) = (𝐹𝑋) ↔ (𝐹𝑦) = (𝐹𝑋)))
87cbvralvw 3383 . . . . 5 (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) ↔ ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
98biimpi 215 . . . 4 (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
10 fveq2 6774 . . . . 5 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
1110iuneqconst 4935 . . . 4 ((𝑋𝑆 ∧ ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
126, 9, 11syl2an 596 . . 3 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
135, 12eqtr3d 2780 . 2 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → (𝐹𝑆) = (𝐹𝑋))
1413ex 413 1 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → (𝐹𝑆) = (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wss 3887   cuni 4839   ciun 4924  cima 5592  Fun wfun 6427  wf 6429  cfv 6433
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441
This theorem is referenced by:  uniimaprimaeqfv  44834
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