Theorem abfmpeld 30393

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 1924	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥{𝑦 ∣ 𝜓} ∈ V)
3		csbexg 5206	. . . . . . . . 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 6765	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓})
7	4, 6	sylan2 594	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓})
8		csbab 4388	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜓} = {𝑦 ∣ [𝐴 / 𝑥]𝜓}
9	7, 8	syl6eq 2872	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜓})
10	9	eleq2d 2898	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓}))
11	10	adantl 484	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓}))
12		simpll 765	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝑉)
13		abfmpeld.3	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))
14	13	ancomsd 468	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)))
15	14	adantl 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)))
16	15	impl 458	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
17	12, 16	sbcied 3813	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝜑) ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
18	17	ex 415	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)))
19	18	alrimiv 1924	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)))
20		elabgt 3662	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓} ↔ 𝜒))
21	19, 20	sylan2 594	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜓} ↔ 𝜒))
22	11, 21	bitrd 281	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∈ 𝑉 ∧ 𝜑)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))
23	22	an13s 649	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒))
24	23	ex 415	1 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  {cab 2799  Vcvv 3494  [wsbc 3771  ⦋csb 3882   ↦ cmpt 5138  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357

This theorem is referenced by: (None)