Theorem fimarab 29725

Description: Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.)

Assertion

Ref	Expression
fimarab	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦})

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem fimarab

Step	Hyp	Ref	Expression
1		nfv 1980	. 2 ⊢ Ⅎ𝑦(𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴)
2		nfcv 2890	. 2 ⊢ Ⅎ𝑦(𝐹 “ 𝑋)
3		nfrab1 3249	. 2 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}
4		ffn 6194	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		fvelimab 6403	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦))
6	5	anbi2d 742	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)))
7	4, 6	sylan 489	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)))
8		imassrn 5623	. . . . . . 7 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
9		frn 6202	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	8, 9	syl5ss 3743	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)
11	10	adantr 472	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) ⊆ 𝐵)
12	11	sseld 3731	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) → 𝑦 ∈ 𝐵))
13	12	pm4.71rd 670	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋))))
14		rabid 3242	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦))
15	14	a1i 11	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)))
16	7, 13, 15	3bitr4d 300	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}))
17	1, 2, 3, 16	eqrd 3751	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1620   ∈ wcel 2127  ∃wrex 3039  {crab 3042   ⊆ wss 3703  ran crn 5255   “ cima 5257   Fn wfn 6032  ⟶wf 6033  ‘cfv 6037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045

This theorem is referenced by:  locfinreflem  30187