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Theorem abfmpel 32666
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 5310 . . . . . 6 𝐴 / 𝑥{𝑦𝜑} ∈ V
3 abfmpel.1 . . . . . . 7 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
43fvmpts 7018 . . . . . 6 ((𝐴𝑉𝐴 / 𝑥{𝑦𝜑} ∈ V) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜑})
52, 4mpan2 691 . . . . 5 (𝐴𝑉 → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜑})
6 csbab 4439 . . . . 5 𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
75, 6eqtrdi 2792 . . . 4 (𝐴𝑉 → (𝐹𝐴) = {𝑦[𝐴 / 𝑥]𝜑})
87eleq2d 2826 . . 3 (𝐴𝑉 → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
98adantr 480 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑}))
10 simpl 482 . . . . . . 7 ((𝐴𝑉𝑦 = 𝐵) → 𝐴𝑉)
11 abfmpel.3 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
1211ancoms 458 . . . . . . . 8 ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜑𝜓))
1312adantll 714 . . . . . . 7 (((𝐴𝑉𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
1410, 13sbcied 3831 . . . . . 6 ((𝐴𝑉𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜑𝜓))
1514ex 412 . . . . 5 (𝐴𝑉 → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓)))
1615alrimiv 1926 . . . 4 (𝐴𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓)))
17 elabgt 3671 . . . 4 ((𝐵𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓))) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
1816, 17sylan2 593 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
1918ancoms 458 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜑} ↔ 𝜓))
209, 19bitrd 279 1 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wcel 2107  {cab 2713  Vcvv 3479  [wsbc 3787  csb 3898  cmpt 5224  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568
This theorem is referenced by:  issiga  34114  ismeas  34201
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