Theorem abfmpeld 32907

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 1950	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥{𝑦 ∣ 𝜓} ∈ V)
3		csbexg 5264	. . . . . . . . 9 ⊢ (∀𝑥{𝑦 ∣ 𝜓} ∈ V → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V)
5		abfmpeld.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜓})
6	5	fvmpts 6983	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓})
7	4, 6	sylan2 604	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓})
8		csbab 4397	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} = {𝑦 ∣ [𝐴 / 𝑥]𝜓}
9	7, 8	eqtrdi 2816	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜓})
10	9	eleq2d 2851	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓}))
11	10	adantl 486	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓}))
12		simpll 778	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝑉)
13		abfmpeld.3	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))
14	13	ancomsd 470	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)))
15	14	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)))
16	15	impl 460	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
17	12, 16	sbcied 3790	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
18	17	ex 417	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)))
19	18	alrimiv 1950	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)))
20		elabgt 3634	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓} ↔ 𝜒))
21	19, 20	sylan2 604	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓} ↔ 𝜒))
22	11, 21	bitrd 282	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))
23	22	an13s 663	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))
24	23	ex 417	1 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563   ∈ wcel 2145  {cab 2743  Vcvv 3457  [wsbc 3747  ⦋csb 3855   ↦ cmpt 5185  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6481  df-fun 6527  df-fv 6533

This theorem is referenced by: (None)