Theorem uniimafveqt 48054

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 6709	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
2	1	3ad2ant1 1149	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → Fun 𝐹)
3	2	adantr 485	. . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → Fun 𝐹)
4		funiunfv 7247	. . . 4 ⊢ (Fun 𝐹 → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑆))
5	3, 4	syl 18	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑆))
6		simp3 1154	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
7		fveqeq2 6891	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑦) = (𝐹‘𝑋)))
8	7	cbvralvw 3249	. . . . 5 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
9	8	biimpi 219	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
10		fveq2 6882	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
11	10	iuneqconst 4970	. . . 4 ⊢ ((𝑋 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
12	6, 9, 11	syl2an 607	. . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
13	5, 12	eqtr3d 2806	. 2 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋)) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋))
14	13	ex 417	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3911  ∪ cuni 4874  ∪ ciun 4958   “ cima 5665  Fun wfun 6531  ⟶wf 6533  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is referenced by:  uniimaprimaeqfv  48055