Theorem abfmpel 32937

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 5273	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V
3		abfmpel.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑})
4	3	fvmpts 6991	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
5	2, 4	mpan2 703	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
6		csbab 4403	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑}
7	5, 6	eqtrdi 2820	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜑})
8	7	eleq2d 2855	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑}))
9	8	adantr 485	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑}))
10		simpl 487	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝑉)
11		abfmpel.3	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
12	11	ancoms 463	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
13	12	adantll 726	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
14	10, 13	sbcied 3796	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
15	14	ex 417	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)))
16	15	alrimiv 1954	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)))
17		elabgt 3640	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
18	16, 17	sylan2 604	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
19	18	ancoms 463	. 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 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149  {cab 2747  Vcvv 3463  [wsbc 3753  ⦋csb 3861   ↦ cmpt 5193  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6535  df-fv 6541

This theorem is referenced by:  issiga  34443  ismeas  34530