Theorem elfvmptrab 7001

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 3871	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → ⦋𝑥 / 𝑚⦌𝑀 = 𝑀)
3	2	eqcomd 2767	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑀 = ⦋𝑥 / 𝑚⦌𝑀)
4		rabeq 3427	. . . . . 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 5194	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
7	1, 6	eqtri 2784	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
8		csbconstg 3871	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 = 𝑀)
9		elfvmptrab.v	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V)
10	8, 9	eqeltrd 2861	. . 3 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
11	7, 10	elfvmptrab1w 6999	. 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
12	8	eleq2d 2847	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 ↔ 𝑌 ∈ 𝑀))
13	12	biimpd 231	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 → 𝑌 ∈ 𝑀))
14	13	imdistani 576	. 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 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453  ⦋csb 3852   ↦ cmpt 5180  ‘cfv 6517

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 5245  ax-nul 5255  ax-pr 5389

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-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fv 6525

This theorem is referenced by: (None)