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Theorem abfmpel 28835
 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 4721 . . . . . 6 𝐴 / 𝑥{𝑦𝜑} ∈ V
3 abfmpel.1 . . . . . . 7 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
43fvmpts 6194 . . . . . 6 ((𝐴𝑉𝐴 / 𝑥{𝑦𝜑} ∈ V) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜑})
52, 4mpan2 703 . . . . 5 (𝐴𝑉 → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜑})
6 csbab 3960 . . . . 5 𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
75, 6syl6eq 2660 . . . 4 (𝐴𝑉 → (𝐹𝐴) = {𝑦[𝐴 / 𝑥]𝜑})
87eleq2d 2673 . . 3 (𝐴𝑉 → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
98adantr 480 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
10 simpl 472 . . . . . . 7 ((𝐴𝑉𝑦 = 𝐵) → 𝐴𝑉)
11 abfmpel.3 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
1211ancoms 468 . . . . . . . 8 ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜑𝜓))
1312adantll 746 . . . . . . 7 (((𝐴𝑉𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
1410, 13sbcied 3439 . . . . . 6 ((𝐴𝑉𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜑𝜓))
1514ex 449 . . . . 5 (𝐴𝑉 → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓)))
1615alrimiv 1842 . . . 4 (𝐴𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓)))
17 elabgt 3316 . . . 4 ((𝐵𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓))) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
1816, 17sylan2 490 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
1918ancoms 468 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
209, 19bitrd 267 1 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402  ⦋csb 3499   ↦ cmpt 4643  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  issiga  29501  ismeas  29589
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