Theorem uniimafveqt 44721

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:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋)))

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.

Step	Hyp	Ref	Expression
1		ffun 6587	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
2	1	3ad2ant1 1131	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → Fun 𝐹)
3	2	adantr 480	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → Fun 𝐹)
4		funiunfv 7103	. . . 4 ⊢ (Fun 𝐹 → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑆))
5	3, 4	syl 17	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑆))
6		simp3 1136	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
7		fveqeq2 6765	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑦) = (𝐹‘𝑋)))
8	7	cbvralvw 3372	. . . . 5 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
9	8	biimpi 215	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
10		fveq2 6756	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
11	10	iuneqconst 4932	. . . 4 ⊢ ((𝑋 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
12	6, 9, 11	syl2an 595	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
13	5, 12	eqtr3d 2780	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋))
14	13	ex 412	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  ∪ cuni 4836  ∪ ciun 4921   “ cima 5583  Fun wfun 6412  ⟶wf 6414  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426

This theorem is referenced by:  uniimaprimaeqfv  44722