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Theorem uniimafveqt 45984
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 6717 . . . . . 6 (𝐹:𝐴𝐵 → Fun 𝐹)
213ad2ant1 1134 . . . . 5 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → Fun 𝐹)
32adantr 482 . . . 4 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → Fun 𝐹)
4 funiunfv 7242 . . . 4 (Fun 𝐹 𝑦𝑆 (𝐹𝑦) = (𝐹𝑆))
53, 4syl 17 . . 3 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑆))
6 simp3 1139 . . . 4 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → 𝑋𝑆)
7 fveqeq2 6897 . . . . . 6 (𝑥 = 𝑦 → ((𝐹𝑥) = (𝐹𝑋) ↔ (𝐹𝑦) = (𝐹𝑋)))
87cbvralvw 3235 . . . . 5 (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) ↔ ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
98biimpi 215 . . . 4 (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
10 fveq2 6888 . . . . 5 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
1110iuneqconst 5007 . . . 4 ((𝑋𝑆 ∧ ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
126, 9, 11syl2an 597 . . 3 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
135, 12eqtr3d 2775 . 2 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → (𝐹𝑆) = (𝐹𝑋))
1413ex 414 1 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → (𝐹𝑆) = (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wss 3947   cuni 4907   ciun 4996  cima 5678  Fun wfun 6534  wf 6536  cfv 6540
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548
This theorem is referenced by:  uniimaprimaeqfv  45985
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