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Theorem uniimafveqt 46349
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 6721 . . . . . 6 (𝐹:𝐴𝐵 → Fun 𝐹)
213ad2ant1 1132 . . . . 5 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → Fun 𝐹)
32adantr 480 . . . 4 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → Fun 𝐹)
4 funiunfv 7250 . . . 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 6901 . . . . . 6 (𝑥 = 𝑦 → ((𝐹𝑥) = (𝐹𝑋) ↔ (𝐹𝑦) = (𝐹𝑋)))
87cbvralvw 3233 . . . . 5 (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) ↔ ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
98biimpi 215 . . . 4 (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
10 fveq2 6892 . . . . 5 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
1110iuneqconst 5009 . . . 4 ((𝑋𝑆 ∧ ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
126, 9, 11syl2an 595 . . 3 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
135, 12eqtr3d 2773 . 2 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → (𝐹𝑆) = (𝐹𝑋))
1413ex 412 1 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → (𝐹𝑆) = (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wss 3949   cuni 4909   ciun 4998  cima 5680  Fun wfun 6538  wf 6540  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552
This theorem is referenced by:  uniimaprimaeqfv  46350
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