Theorem abfmpeld 29454

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 1855	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥{𝑦 ∣ 𝜓} ∈ V)
3		csbexg 4792	. . . . . . . . 9 ⊢ (∀𝑥{𝑦 ∣ 𝜓} ∈ V → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V)
5		abfmpeld.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜓})
6	5	fvmpts 6285	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓})
7	4, 6	sylan2 491	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓})
8		csbab 4008	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} = {𝑦 ∣ [𝐴 / 𝑥]𝜓}
9	7, 8	syl6eq 2672	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜓})
10	9	eleq2d 2687	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓}))
11	10	adantl 482	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓}))
12		simpll 790	. . . . . . . 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 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)))
16	15	impl 650	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
17	12, 16	sbcied 3472	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
18	17	ex 450	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)))
19	18	alrimiv 1855	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)))
20		elabgt 3347	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓} ↔ 𝜒))
21	19, 20	sylan2 491	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓} ↔ 𝜒))
22	11, 21	bitrd 268	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))
23	22	an13s 845	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))
24	23	ex 450	1 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200  [wsbc 3435  ⦋csb 3533   ↦ cmpt 4729  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by: (None)