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Theorem fimarab 30398
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 1915 . 2 𝑦(𝐹:𝐴𝐵𝑋𝐴)
2 nfcv 2979 . 2 𝑦(𝐹𝑋)
3 nfrab1 3365 . 2 𝑦{𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦}
4 ffn 6494 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 fvelimab 6719 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ ∃𝑥𝑋 (𝐹𝑥) = 𝑦))
65anbi2d 631 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝑦𝐵𝑦 ∈ (𝐹𝑋)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
74, 6sylan 583 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → ((𝑦𝐵𝑦 ∈ (𝐹𝑋)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
8 fimass 6536 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ 𝐵)
98adantr 484 . . . . 5 ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) ⊆ 𝐵)
109sseld 3941 . . . 4 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) → 𝑦𝐵))
1110pm4.71rd 566 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ (𝑦𝐵𝑦 ∈ (𝐹𝑋))))
12 rabid 3359 . . . 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 314 . 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 209  wa 399   = wceq 1538  wcel 2114  wrex 3131  {crab 3134  wss 3908  cima 5535   Fn wfn 6329  wf 6330  cfv 6334
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342
This theorem is referenced by:  locfinreflem  31162
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