Theorem dfaimafn2 48344

Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6953. (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 48343	. . 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 2813	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
4		df-sn 4585	. . . . 5 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ 𝑦 = (𝐹'''𝑥)}
5		eqcom 2767	. . . . . 6 ⊢ (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
6	5	abbii 2827	. . . . 5 ⊢ {𝑦 ∣ 𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
7	4, 6	eqtri 2783	. . . 4 ⊢ {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
8	7	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
9	8	iuneq2i 4972	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
10	3, 9	eqtr4di 2813	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 {(𝐹'''𝑥)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ⊆ wss 3896  {csn 4584  ∪ ciun 4950  dom cdm 5651   “ cima 5654  Fun wfun 6537  '''cafv 48296

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-fv 6551  df-aiota 48263  df-dfat 48298  df-afv 48299

This theorem is used by: (None)