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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: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 1980 . 2 𝑦(𝐹:𝐴𝐵𝑋𝐴)
2 nfcv 2890 . 2 𝑦(𝐹𝑋)
3 nfrab1 3249 . 2 𝑦{𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦}
4 ffn 6194 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 fvelimab 6403 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ ∃𝑥𝑋 (𝐹𝑥) = 𝑦))
65anbi2d 742 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝑦𝐵𝑦 ∈ (𝐹𝑋)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
74, 6sylan 489 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → ((𝑦𝐵𝑦 ∈ (𝐹𝑋)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
8 imassrn 5623 . . . . . . 7 (𝐹𝑋) ⊆ ran 𝐹
9 frn 6202 . . . . . . 7 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
108, 9syl5ss 3743 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ 𝐵)
1110adantr 472 . . . . 5 ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) ⊆ 𝐵)
1211sseld 3731 . . . 4 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) → 𝑦𝐵))
1312pm4.71rd 670 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ (𝑦𝐵𝑦 ∈ (𝐹𝑋))))
14 rabid 3242 . . . 4 (𝑦 ∈ {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦} ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦))
1514a1i 11 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦} ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
167, 13, 153bitr4d 300 . 2 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ 𝑦 ∈ {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦}))
171, 2, 3, 16eqrd 3751 1 ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) = {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦})
Colors of variables: wff setvar class
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
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