Theorem uniimafveqt 43615

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 6514	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
2	1	3ad2ant1 1128	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → Fun 𝐹)
3	2	adantr 483	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → Fun 𝐹)
4		funiunfv 7004	. . . 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 1133	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
7		fveqeq2 6676	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑦) = (𝐹‘𝑋)))
8	7	cbvralvw 3448	. . . . 5 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
9	8	biimpi 218	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
10		fveq2 6667	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
11	10	iuneqconst 4927	. . . 4 ⊢ ((𝑋 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
12	6, 9, 11	syl2an 597	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
13	5, 12	eqtr3d 2857	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋))
14	13	ex 415	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113  ∀wral 3137   ⊆ wss 3933  ∪ cuni 4835  ∪ ciun 4916   “ cima 5555  Fun wfun 6346  ⟶wf 6348  ‘cfv 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3495  df-sbc 3771  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-iun 4918  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5457  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360

This theorem is referenced by:  uniimaprimaeqfv  43616