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Theorem abfmpeld 32671
Description: Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
Hypotheses
Ref Expression
abfmpeld.1 𝐹 = (𝑥𝑉 ↦ {𝑦𝜓})
abfmpeld.2 (𝜑 → {𝑦𝜓} ∈ V)
abfmpeld.3 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
Assertion
Ref Expression
abfmpeld (𝜑 → ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝑦,𝑊   𝜒,𝑥,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑊(𝑥)

Proof of Theorem abfmpeld
StepHypRef Expression
1 abfmpeld.2 . . . . . . . . . 10 (𝜑 → {𝑦𝜓} ∈ V)
21alrimiv 1925 . . . . . . . . 9 (𝜑 → ∀𝑥{𝑦𝜓} ∈ V)
3 csbexg 5316 . . . . . . . . 9 (∀𝑥{𝑦𝜓} ∈ V → 𝐴 / 𝑥{𝑦𝜓} ∈ V)
42, 3syl 17 . . . . . . . 8 (𝜑𝐴 / 𝑥{𝑦𝜓} ∈ V)
5 abfmpeld.1 . . . . . . . . 9 𝐹 = (𝑥𝑉 ↦ {𝑦𝜓})
65fvmpts 7019 . . . . . . . 8 ((𝐴𝑉𝐴 / 𝑥{𝑦𝜓} ∈ V) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜓})
74, 6sylan2 593 . . . . . . 7 ((𝐴𝑉𝜑) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜓})
8 csbab 4446 . . . . . . 7 𝐴 / 𝑥{𝑦𝜓} = {𝑦[𝐴 / 𝑥]𝜓}
97, 8eqtrdi 2791 . . . . . 6 ((𝐴𝑉𝜑) → (𝐹𝐴) = {𝑦[𝐴 / 𝑥]𝜓})
109eleq2d 2825 . . . . 5 ((𝐴𝑉𝜑) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓}))
1110adantl 481 . . . 4 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓}))
12 simpll 767 . . . . . . . 8 (((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) → 𝐴𝑉)
13 abfmpeld.3 . . . . . . . . . . 11 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
1413ancomsd 465 . . . . . . . . . 10 (𝜑 → ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜓𝜒)))
1514adantl 481 . . . . . . . . 9 ((𝐴𝑉𝜑) → ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜓𝜒)))
1615impl 455 . . . . . . . 8 ((((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
1712, 16sbcied 3837 . . . . . . 7 (((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓𝜒))
1817ex 412 . . . . . 6 ((𝐴𝑉𝜑) → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒)))
1918alrimiv 1925 . . . . 5 ((𝐴𝑉𝜑) → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒)))
20 elabgt 3672 . . . . 5 ((𝐵𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒))) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓} ↔ 𝜒))
2119, 20sylan2 593 . . . 4 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓} ↔ 𝜒))
2211, 21bitrd 279 . . 3 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒))
2322an13s 651 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒))
2423ex 412 1 (𝜑 → ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  [wsbc 3791  csb 3908  cmpt 5231  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by: (None)
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