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Theorem abfmpeld 29296
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 1852 . . . . . . . . 9 (𝜑 → ∀𝑥{𝑦𝜓} ∈ V)
3 csbexg 4752 . . . . . . . . 9 (∀𝑥{𝑦𝜓} ∈ V → 𝐴 / 𝑥{𝑦𝜓} ∈ V)
42, 3syl 17 . . . . . . . 8 (𝜑𝐴 / 𝑥{𝑦𝜓} ∈ V)
5 abfmpeld.1 . . . . . . . . 9 𝐹 = (𝑥𝑉 ↦ {𝑦𝜓})
65fvmpts 6242 . . . . . . . 8 ((𝐴𝑉𝐴 / 𝑥{𝑦𝜓} ∈ V) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜓})
74, 6sylan2 491 . . . . . . 7 ((𝐴𝑉𝜑) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜓})
8 csbab 3980 . . . . . . 7 𝐴 / 𝑥{𝑦𝜓} = {𝑦[𝐴 / 𝑥]𝜓}
97, 8syl6eq 2671 . . . . . 6 ((𝐴𝑉𝜑) → (𝐹𝐴) = {𝑦[𝐴 / 𝑥]𝜓})
109eleq2d 2684 . . . . 5 ((𝐴𝑉𝜑) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓}))
1110adantl 482 . . . 4 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓}))
12 simpll 789 . . . . . . . 8 (((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) → 𝐴𝑉)
13 abfmpeld.3 . . . . . . . . . . 11 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
1413ancomsd 470 . . . . . . . . . 10 (𝜑 → ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜓𝜒)))
1514adantl 482 . . . . . . . . 9 ((𝐴𝑉𝜑) → ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜓𝜒)))
1615impl 649 . . . . . . . 8 ((((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
1712, 16sbcied 3454 . . . . . . 7 (((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓𝜒))
1817ex 450 . . . . . 6 ((𝐴𝑉𝜑) → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒)))
1918alrimiv 1852 . . . . 5 ((𝐴𝑉𝜑) → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒)))
20 elabgt 3330 . . . . 5 ((𝐵𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒))) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓} ↔ 𝜒))
2119, 20sylan2 491 . . . 4 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓} ↔ 𝜒))
2211, 21bitrd 268 . . 3 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒))
2322an13s 844 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒))
2423ex 450 1 (𝜑 → ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3186  [wsbc 3417  csb 3514  cmpt 4673  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855
This theorem is referenced by: (None)
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