Theorem elfvmptrab 6806

Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Hypotheses

Ref	Expression
elfvmptrab.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑})
elfvmptrab.v	⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V)

Assertion

Ref	Expression
elfvmptrab	⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀))

Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑉   𝑥,𝑋,𝑦   𝑦,𝑌

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑦)   𝑌(𝑥)

Proof of Theorem elfvmptrab

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvmptrab.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑})
2		csbconstg 3810	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → ⦋𝑥 / 𝑚⦌𝑀 = 𝑀)
3	2	eqcomd 2745	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑀 = ⦋𝑥 / 𝑚⦌𝑀)
4		rabeq 3386	. . . . . 6 ⊢ (𝑀 = ⦋𝑥 / 𝑚⦌𝑀 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
5	3, 4	syl 17	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
6	5	mpteq2ia 5122	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
7	1, 6	eqtri 2762	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
8		csbconstg 3810	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 = 𝑀)
9		elfvmptrab.v	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V)
10	8, 9	eqeltrd 2834	. . 3 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
11	7, 10	elfvmptrab1w 6804	. 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
12	8	eleq2d 2819	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 ↔ 𝑌 ∈ 𝑀))
13	12	biimpd 232	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 → 𝑌 ∈ 𝑀))
14	13	imdistani 572	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀))
15	11, 14	syl 17	1 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114  {crab 3058  Vcvv 3399  ⦋csb 3791   ↦ cmpt 5111  ‘cfv 6340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fv 6348

This theorem is referenced by: (None)