Theorem fimarab 30390

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 1915	. 2 ⊢ Ⅎ𝑦(𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴)
2		nfcv 2977	. 2 ⊢ Ⅎ𝑦(𝐹 “ 𝑋)
3		nfrab1 3384	. 2 ⊢ Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}
4		ffn 6514	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
5		fvelimab 6737	. . . . 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 582	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦)))
8		fimass 6555	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)
9	8	adantr 483	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) ⊆ 𝐵)
10	9	sseld 3966	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) → 𝑦 ∈ 𝐵))
11	10	pm4.71rd 565	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ (𝐹 “ 𝑋))))
12		rabid 3378	. . . 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 313	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝑋) ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦}))
15	1, 2, 3, 14	eqrd 3986	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐹 “ 𝑋) = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = 𝑦})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  {crab 3142   ⊆ wss 3936   “ cima 5558   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363

This theorem is referenced by:  locfinreflem  31104