Theorem abfmpel 32672

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

Assertion

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

Step	Hyp	Ref	Expression
1		abfmpel.2	. . . . . . 7 ⊢ {𝑦 ∣ 𝜑} ∈ V
2	1	csbex 5317	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V
3		abfmpel.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑})
4	3	fvmpts 7019	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
5	2, 4	mpan2 691	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
6		csbab 4446	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑}
7	5, 6	eqtrdi 2791	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜑})
8	7	eleq2d 2825	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑}))
9	8	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑}))
10		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝑉)
11		abfmpel.3	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
12	11	ancoms 458	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
13	12	adantll 714	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
14	10, 13	sbcied 3837	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
15	14	ex 412	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)))
16	15	alrimiv 1925	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)))
17		elabgt 3672	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
18	16, 17	sylan2 593	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
19	18	ancoms 458	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
20	9, 19	bitrd 279	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜓))

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:  issiga  34093  ismeas  34180