Theorem dfimafn2 6942

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 6941	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})
2		iunab 5027	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}
3	1, 2	eqtr4di 2788	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
4		df-sn 4602	. . . . 5 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ 𝑦 = (𝐹‘𝑥)}
5		eqcom 2742	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
6	5	abbii 2802	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
7	4, 6	eqtri 2758	. . . 4 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
8	7	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
9	8	iuneq2i 4989	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
10	3, 9	eqtr4di 2788	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∃wrex 3060   ⊆ wss 3926  {csn 4601  ∪ ciun 4967  dom cdm 5654   “ cima 5657  Fun wfun 6525  ‘cfv 6531

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is referenced by:  uniiccdif  25531