Theorem dfimafn2 5606

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 5605	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})
2		iunab 3959	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}
3	1, 2	eqtr4di 2244	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
4		df-sn 3624	. . . . 5 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ 𝑦 = (𝐹‘𝑥)}
5		eqcom 2195	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
6	5	abbii 2309	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
7	4, 6	eqtri 2214	. . . 4 ⊢ {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
8	7	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹‘𝑥)} = {𝑦 ∣ (𝐹‘𝑥) = 𝑦})
9	8	iuneq2i 3930	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹‘𝑥) = 𝑦}
10	3, 9	eqtr4di 2244	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹‘𝑥)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473   ⊆ wss 3153  {csn 3618  ∪ ciun 3912  dom cdm 4659   “ cima 4662  Fun wfun 5248  ‘cfv 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by: (None)