Theorem abfmpeld 30991

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 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜓})
abfmpeld.2	⊢ (𝜑 → {𝑦 ∣ 𝜓} ∈ V)
abfmpeld.3	⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
abfmpeld	⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 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

Step	Hyp	Ref	Expression
1		abfmpeld.2	. . . . . . . . . 10 ⊢ (𝜑 → {𝑦 ∣ 𝜓} ∈ V)
2	1	alrimiv 1930	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥{𝑦 ∣ 𝜓} ∈ V)
3		csbexg 5234	. . . . . . . . 9 ⊢ (∀𝑥{𝑦 ∣ 𝜓} ∈ V → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V)
5		abfmpeld.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜓})
6	5	fvmpts 6878	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓})
7	4, 6	sylan2 593	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓})
8		csbab 4371	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} = {𝑦 ∣ [𝐴 / 𝑥]𝜓}
9	7, 8	eqtrdi 2794	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜓})
10	9	eleq2d 2824	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓}))
11	10	adantl 482	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓}))
12		simpll 764	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝑉)
13		abfmpeld.3	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))
14	13	ancomsd 466	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)))
15	14	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)))
16	15	impl 456	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
17	12, 16	sbcied 3761	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
18	17	ex 413	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)))
19	18	alrimiv 1930	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)))
20		elabgt 3603	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓} ↔ 𝜒))
21	19, 20	sylan2 593	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓} ↔ 𝜒))
22	11, 21	bitrd 278	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))
23	22	an13s 648	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))
24	23	ex 413	1 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  Vcvv 3432  [wsbc 3716  ⦋csb 3832   ↦ cmpt 5157  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441

This theorem is referenced by: (None)