Theorem abfmpel 30418

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 5179	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V
3		abfmpel.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑})
4	3	fvmpts 6748	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
5	2, 4	mpan2 690	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
6		csbab 4345	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑}
7	5, 6	eqtrdi 2849	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜑})
8	7	eleq2d 2875	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑}))
9	8	adantr 484	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑}))
10		simpl 486	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝑉)
11		abfmpel.3	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
12	11	ancoms 462	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
13	12	adantll 713	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
14	10, 13	sbcied 3762	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
15	14	ex 416	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)))
16	15	alrimiv 1928	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)))
17		elabgt 3609	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
18	16, 17	sylan2 595	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
19	18	ancoms 462	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
20	9, 19	bitrd 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2776  Vcvv 3441  [wsbc 3720  ⦋csb 3828   ↦ cmpt 5110  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332

This theorem is referenced by:  issiga  31481  ismeas  31568