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Theorem uniimafveqt 44367
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 6507 . . . . . 6 (𝐹:𝐴𝐵 → Fun 𝐹)
213ad2ant1 1134 . . . . 5 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → Fun 𝐹)
32adantr 484 . . . 4 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → Fun 𝐹)
4 funiunfv 7018 . . . 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 6683 . . . . . 6 (𝑥 = 𝑦 → ((𝐹𝑥) = (𝐹𝑋) ↔ (𝐹𝑦) = (𝐹𝑋)))
87cbvralvw 3349 . . . . 5 (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) ↔ ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
98biimpi 219 . . . 4 (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
10 fveq2 6674 . . . . 5 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
1110iuneqconst 4892 . . . 4 ((𝑋𝑆 ∧ ∀𝑦𝑆 (𝐹𝑦) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
126, 9, 11syl2an 599 . . 3 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → 𝑦𝑆 (𝐹𝑦) = (𝐹𝑋))
135, 12eqtr3d 2775 . 2 (((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) ∧ ∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋)) → (𝐹𝑆) = (𝐹𝑋))
1413ex 416 1 ((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → (𝐹𝑆) = (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  wss 3843   cuni 4796   ciun 4881  cima 5528  Fun wfun 6333  wf 6335  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347
This theorem is referenced by:  uniimaprimaeqfv  44368
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