Theorem dfaimafn2 47523

Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6905. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
dfaimafn2	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfaimafn2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfaimafn 47522	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦})
2		iunab 5009	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}
3	1, 2	eqtr4di 2790	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
4		df-sn 4583	. . . . 5 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ 𝑦 = (𝐹'''𝑥)}
5		eqcom 2744	. . . . . 6 ⊢ (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
6	5	abbii 2804	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
7	4, 6	eqtri 2760	. . . 4 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
8	7	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
9	8	iuneq2i 4970	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
10	3, 9	eqtr4di 2790	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ⊆ wss 3903  {csn 4582  ∪ ciun 4948  dom cdm 5632   “ cima 5635  Fun wfun 6494  '''cafv 47474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-aiota 47442  df-dfat 47476  df-afv 47477

This theorem is referenced by: (None)