Theorem uniimafveqt 47255

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 6750	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
2	1	3ad2ant1 1133	. . . . 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 7285	. . . 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 1138	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
7		fveqeq2 6929	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑦) = (𝐹‘𝑋)))
8	7	cbvralvw 3243	. . . . 5 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
9	8	biimpi 216	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∀𝑦 ∈ 𝑆 (𝐹‘𝑦) = (𝐹‘𝑋))
10		fveq2 6920	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
11	10	iuneqconst 5026	. . . 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 2782	. 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  ∪ cuni 4931  ∪ ciun 5015   “ cima 5703  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  uniimaprimaeqfv  47256