Theorem fimarab 6982

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 1911	. 2 ⊢ Ⅎ𝑦(𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴)
2		nfcv 2902	. 2 ⊢ Ⅎ𝑦(𝐹 “ 𝑋)
3		nfrab1 3453	. 2 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}
4		ffn 6736	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		fvelimab 6980	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦))
6	5	anbi2d 630	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)))
7	4, 6	sylan 580	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)))
8		fimass 6756	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)
9	8	adantr 480	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) ⊆ 𝐵)
10	9	sseld 3993	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) → 𝑦 ∈ 𝐵))
11	10	pm4.71rd 562	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋))))
12		rabid 3454	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦))
13	12	a1i 11	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦} ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)))
14	7, 11, 13	3bitr4d 311	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}))
15	1, 2, 3, 14	eqrd 4014	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∃wrex 3067  {crab 3432   ⊆ wss 3962   “ cima 5691   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570

This theorem is referenced by:  locfinreflem  33800  uspgrlimlem1  47890  uspgrlimlem2  47891