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Theorem abfmpel 29583
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 4826 . . . . . 6 𝐴 / 𝑥{𝑦𝜑} ∈ V
3 abfmpel.1 . . . . . . 7 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
43fvmpts 6324 . . . . . 6 ((𝐴𝑉𝐴 / 𝑥{𝑦𝜑} ∈ V) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜑})
52, 4mpan2 707 . . . . 5 (𝐴𝑉 → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜑})
6 csbab 4041 . . . . 5 𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
75, 6syl6eq 2701 . . . 4 (𝐴𝑉 → (𝐹𝐴) = {𝑦[𝐴 / 𝑥]𝜑})
87eleq2d 2716 . . 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 750 . . . . . . 7 (((𝐴𝑉𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
1410, 13sbcied 3505 . . . . . 6 ((𝐴𝑉𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜑𝜓))
1514ex 449 . . . . 5 (𝐴𝑉 → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓)))
1615alrimiv 1895 . . . 4 (𝐴𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑𝜓)))
17 elabgt 3379 . . . 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 268 1 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231  [wsbc 3468  csb 3566  cmpt 4762  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  issiga  30302  ismeas  30390
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