Theorem dfimafn2 6924

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 6923	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})
2		iunab 5015	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}
3	1, 2	eqtr4di 2782	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
4		df-sn 4590	. . . . 5 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ 𝑦 = (𝐹‘𝑥)}
5		eqcom 2736	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
6	5	abbii 2796	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
7	4, 6	eqtri 2752	. . . 4 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
8	7	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
9	8	iuneq2i 4977	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
10	3, 9	eqtr4di 2782	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ⊆ wss 3914  {csn 4589  ∪ ciun 4955  dom cdm 5638   “ cima 5641  Fun wfun 6505  ‘cfv 6511

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  uniiccdif  25479