Theorem abfmpel 32818

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 5258	. . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V
3		abfmpel.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑})
4	3	fvmpts 6974	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
5	2, 4	mpan2 701	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
6		csbab 4391	. . . . 5 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑}
7	5, 6	eqtrdi 2812	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜑})
8	7	eleq2d 2847	. . 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 724	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
14	10, 13	sbcied 3785	. . . . . 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 1946	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)))
17		elabgt 3630	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓))
18	16, 17	sylan2 602	. . 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 281	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  {cab 2739  Vcvv 3453  [wsbc 3742  ⦋csb 3850   ↦ cmpt 5178  ‘cfv 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524

This theorem is referenced by:  issiga  34370  ismeas  34457