Theorem dfimafn2 5752

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 5751	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})
2		iunab 4059	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}
3	1, 2	eqtr4di 2289	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
4		df-sn 3715	. . . . 5 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ 𝑦 = (𝐹‘𝑥)}
5		eqcom 2240	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
6	5	abbii 2354	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
7	4, 6	eqtri 2259	. . . 4 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
8	7	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
9	8	iuneq2i 4030	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
10	3, 9	eqtr4di 2289	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529   ⊆ wss 3220  {csn 3709  ∪ ciun 4012  dom cdm 4774   “ cima 4777  Fun wfun 5371  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by: (None)