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Theorem fimarab 31404
Description: Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.)
Assertion
Ref Expression
fimarab ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) = {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 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
StepHypRef Expression
1 nfv 1917 . 2 𝑦(𝐹:𝐴𝐵𝑋𝐴)
2 nfcv 2905 . 2 𝑦(𝐹𝑋)
3 nfrab1 3424 . 2 𝑦{𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦}
4 ffn 6665 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 fvelimab 6911 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ ∃𝑥𝑋 (𝐹𝑥) = 𝑦))
65anbi2d 629 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝑦𝐵𝑦 ∈ (𝐹𝑋)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
74, 6sylan 580 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → ((𝑦𝐵𝑦 ∈ (𝐹𝑋)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
8 fimass 6686 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ 𝐵)
98adantr 481 . . . . 5 ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) ⊆ 𝐵)
109sseld 3941 . . . 4 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) → 𝑦𝐵))
1110pm4.71rd 563 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ (𝑦𝐵𝑦 ∈ (𝐹𝑋))))
12 rabid 3425 . . . 4 (𝑦 ∈ {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦} ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦))
1312a1i 11 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦} ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
147, 11, 133bitr4d 310 . 2 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ 𝑦 ∈ {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦}))
151, 2, 3, 14eqrd 3961 1 ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) = {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3071  {crab 3405  wss 3908  cima 5634   Fn wfn 6488  wf 6489  cfv 6493
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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501
This theorem is referenced by:  locfinreflem  32249
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