Theorem elfvmptrab1 7019

Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Usage of this theorem is discouraged because it depends on ax-13 2403. Use the weaker elfvmptrab1w 7018 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elfvmptrab1.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
elfvmptrab1.v	⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)

Assertion

Ref	Expression
elfvmptrab1	⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))

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

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

Proof of Theorem elfvmptrab1

Step	Hyp	Ref	Expression
1		ne0i 4290	. . 3 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝐹‘𝑋) ≠ ∅)
2		ndmfv 6914	. . . 4 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅)
3	2	necon1ai 2984	. . 3 ⊢ ((𝐹‘𝑋) ≠ ∅ → 𝑋 ∈ dom 𝐹)
4		elfvmptrab1.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
5	4	dmmptss 6241	. . . . . . 7 ⊢ dom 𝐹 ⊆ 𝑉
6	5	sseli 3930	. . . . . 6 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉)
7		elfvmptrab1.v	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
8		rabexg 5306	. . . . . . 7 ⊢ (⦋𝑋 / 𝑚⦌𝑀 ∈ V → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V)
9	6, 7, 8	3syl 19	. . . . . 6 ⊢ (𝑋 ∈ dom 𝐹 → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V)
10		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
11		nfsbc1v 3762	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑋 / 𝑥]𝜑
12		nfcv 2924	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑀
13	10, 12	nfcsb 3877	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀
14	11, 13	nfrabw 3450	. . . . . . 7 ⊢ Ⅎ𝑥{𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}
15		csbeq1 3853	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
16		sbceq1a 3753	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ [𝑋 / 𝑥]𝜑))
17	15, 16	rabeqbidv 3432	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
18	10, 14, 17, 4	fvmptf 7012	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
19	6, 9, 18	syl2anc 596	. . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
20	19	eleq2d 2848	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}))
21		elrabi 3644	. . . . . 6 ⊢ (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)
22	6, 21	anim12i 625	. . . . 5 ⊢ ((𝑋 ∈ dom 𝐹 ∧ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
23	22	ex 418	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
24	20, 23	sylbid 243	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
25	1, 3, 24	3syl 19	. 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
26	25	pm2.43i 53	1 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  {crab 3414  Vcvv 3453  [wsbc 3742  ⦋csb 3850  ∅c0 4282   ↦ cmpt 5190  dom cdm 5659  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2403  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545

This theorem is used by: (None)