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Theorem abfmpel 30741
Description: Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
Hypotheses
Ref Expression
abfmpel.1 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
abfmpel.2 {𝑦𝜑} ∈ V
abfmpel.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
abfmpel ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝑦,𝑊   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑊(𝑥)

Proof of Theorem abfmpel
StepHypRef Expression
1 abfmpel.2 . . . . . . 7 {𝑦𝜑} ∈ V
21csbex 5220 . . . . . 6 𝐴 / 𝑥{𝑦𝜑} ∈ V
3 abfmpel.1 . . . . . . 7 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
43fvmpts 6842 . . . . . 6 ((𝐴𝑉𝐴 / 𝑥{𝑦𝜑} ∈ V) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜑})
52, 4mpan2 691 . . . . 5 (𝐴𝑉 → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜑})
6 csbab 4368 . . . . 5 𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
75, 6eqtrdi 2796 . . . 4 (𝐴𝑉 → (𝐹𝐴) = {𝑦[𝐴 / 𝑥]𝜑})
87eleq2d 2825 . . 3 (𝐴𝑉 → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
98adantr 484 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
10 simpl 486 . . . . . . 7 ((𝐴𝑉𝑦 = 𝐵) → 𝐴𝑉)
11 abfmpel.3 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
1211ancoms 462 . . . . . . . 8 ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜑𝜓))
1312adantll 714 . . . . . . 7 (((𝐴𝑉𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
1410, 13sbcied 3755 . . . . . 6 ((𝐴𝑉𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜑𝜓))
1514ex 416 . . . . 5 (𝐴𝑉 → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓)))
1615alrimiv 1935 . . . 4 (𝐴𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓)))
17 elabgt 3595 . . . 4 ((𝐵𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓))) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
1816, 17sylan2 596 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
1918ancoms 462 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
209, 19bitrd 282 1 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wcel 2112  {cab 2716  Vcvv 3422  [wsbc 3710  csb 3827  cmpt 5151  cfv 6400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3711  df-csb 3828  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-mpt 5152  df-id 5471  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-iota 6358  df-fun 6402  df-fv 6408
This theorem is referenced by:  issiga  31821  ismeas  31908
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