Theorem dfimafn2 6815

Description: Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
dfimafn2	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfimafn2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfimafn 6814	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})
2		iunab 4977	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}
3	1, 2	eqtr4di 2797	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
4		df-sn 4559	. . . . 5 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ 𝑦 = (𝐹‘𝑥)}
5		eqcom 2745	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
6	5	abbii 2809	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
7	4, 6	eqtri 2766	. . . 4 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
8	7	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
9	8	iuneq2i 4942	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
10	3, 9	eqtr4di 2797	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064   ⊆ wss 3883  {csn 4558  ∪ ciun 4921  dom cdm 5580   “ cima 5583  Fun wfun 6412  ‘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-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-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-fv 6426

This theorem is referenced by:  uniiccdif  24647