Theorem fimarab 32123

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 1917	. 2 ⊢ Ⅎ𝑦(𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴)
2		nfcv 2903	. 2 ⊢ Ⅎ𝑦(𝐹 “ 𝑋)
3		nfrab1 3451	. 2 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}
4		ffn 6717	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		fvelimab 6964	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦))
6	5	anbi2d 629	. . . 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 6738	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)
9	8	adantr 481	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) ⊆ 𝐵)
10	9	sseld 3981	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) → 𝑦 ∈ 𝐵))
11	10	pm4.71rd 563	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋))))
12		rabid 3452	. . . 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 310	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}))
15	1, 2, 3, 14	eqrd 4001	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432   ⊆ wss 3948   “ cima 5679   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551

This theorem is referenced by:  locfinreflem  33106