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Theorem fimarab 29284
 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 1840 . 2 𝑦(𝐹:𝐴𝐵𝑋𝐴)
2 nfcv 2761 . 2 𝑦(𝐹𝑋)
3 nfrab1 3111 . 2 𝑦{𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦}
4 ffn 6002 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5 fvelimab 6210 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ ∃𝑥𝑋 (𝐹𝑥) = 𝑦))
65anbi2d 739 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝑦𝐵𝑦 ∈ (𝐹𝑋)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
74, 6sylan 488 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → ((𝑦𝐵𝑦 ∈ (𝐹𝑋)) ↔ (𝑦𝐵 ∧ ∃𝑥𝑋 (𝐹𝑥) = 𝑦)))
8 imassrn 5436 . . . . . . 7 (𝐹𝑋) ⊆ ran 𝐹
9 frn 6010 . . . . . . 7 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
108, 9syl5ss 3594 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝑋) ⊆ 𝐵)
1110adantr 481 . . . . 5 ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) ⊆ 𝐵)
1211sseld 3582 . . . 4 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) → 𝑦𝐵))
1312pm4.71rd 666 . . 3 ((𝐹:𝐴𝐵𝑋𝐴) → (𝑦 ∈ (𝐹𝑋) ↔ (𝑦𝐵𝑦 ∈ (𝐹𝑋))))
14 rabid 3106 . . . 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 3602 1 ((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) = {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  {crab 2911   ⊆ wss 3555  ran crn 5075   “ cima 5077   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855 This theorem is referenced by:  locfinreflem  29686
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