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Theorem abfmpeld 29794
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 2007 . . . . . . . . 9 (𝜑 → ∀𝑥{𝑦𝜓} ∈ V)
3 csbexg 4927 . . . . . . . . 9 (∀𝑥{𝑦𝜓} ∈ V → 𝐴 / 𝑥{𝑦𝜓} ∈ V)
42, 3syl 17 . . . . . . . 8 (𝜑𝐴 / 𝑥{𝑦𝜓} ∈ V)
5 abfmpeld.1 . . . . . . . . 9 𝐹 = (𝑥𝑉 ↦ {𝑦𝜓})
65fvmpts 6429 . . . . . . . 8 ((𝐴𝑉𝐴 / 𝑥{𝑦𝜓} ∈ V) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜓})
74, 6sylan2 580 . . . . . . 7 ((𝐴𝑉𝜑) → (𝐹𝐴) = 𝐴 / 𝑥{𝑦𝜓})
8 csbab 4153 . . . . . . 7 𝐴 / 𝑥{𝑦𝜓} = {𝑦[𝐴 / 𝑥]𝜓}
97, 8syl6eq 2821 . . . . . 6 ((𝐴𝑉𝜑) → (𝐹𝐴) = {𝑦[𝐴 / 𝑥]𝜓})
109eleq2d 2836 . . . . 5 ((𝐴𝑉𝜑) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓}))
1110adantl 467 . . . 4 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓}))
12 simpll 750 . . . . . . . 8 (((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) → 𝐴𝑉)
13 abfmpeld.3 . . . . . . . . . . 11 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
1413ancomsd 451 . . . . . . . . . 10 (𝜑 → ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜓𝜒)))
1514adantl 467 . . . . . . . . 9 ((𝐴𝑉𝜑) → ((𝑦 = 𝐵𝑥 = 𝐴) → (𝜓𝜒)))
1615impl 443 . . . . . . . 8 ((((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
1712, 16sbcied 3624 . . . . . . 7 (((𝐴𝑉𝜑) ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓𝜒))
1817ex 397 . . . . . 6 ((𝐴𝑉𝜑) → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒)))
1918alrimiv 2007 . . . . 5 ((𝐴𝑉𝜑) → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒)))
20 elabgt 3498 . . . . 5 ((𝐵𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓𝜒))) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓} ↔ 𝜒))
2119, 20sylan2 580 . . . 4 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ {𝑦[𝐴 / 𝑥]𝜓} ↔ 𝜒))
2211, 21bitrd 268 . . 3 ((𝐵𝑊 ∧ (𝐴𝑉𝜑)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒))
2322an13s 630 . 2 ((𝜑 ∧ (𝐴𝑉𝐵𝑊)) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒))
2423ex 397 1 (𝜑 → ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wcel 2145  {cab 2757  Vcvv 3351  [wsbc 3587  csb 3682  cmpt 4864  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038
This theorem is referenced by: (None)
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