Theorem elfvmptrab1 5771

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. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

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		elfvmptrab1.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
2	1	funmpt2 5390	. . . 4 ⊢ Fun 𝐹
3		funrel 5368	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
4	2, 3	ax-mp 5	. . 3 ⊢ Rel 𝐹
5		relelfvdm 5701	. . 3 ⊢ ((Rel 𝐹 ∧ 𝑌 ∈ (𝐹‘𝑋)) → 𝑋 ∈ dom 𝐹)
6	4, 5	mpan 424	. 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → 𝑋 ∈ dom 𝐹)
7	1	dmmptss 5258	. . . . . 6 ⊢ dom 𝐹 ⊆ 𝑉
8	7	sseli 3233	. . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉)
9		elfvmptrab1.v	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
10		rabexg 4254	. . . . . 6 ⊢ (⦋𝑋 / 𝑚⦌𝑀 ∈ V → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V)
11	8, 9, 10	3syl 17	. . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V)
12		nfcv 2384	. . . . . 6 ⊢ Ⅎ𝑥𝑋
13		nfsbc1v 3060	. . . . . . 7 ⊢ Ⅎ𝑥[𝑋 / 𝑥]𝜑
14		nfcv 2384	. . . . . . . 8 ⊢ Ⅎ𝑥𝑀
15	12, 14	nfcsb 3175	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀
16	13, 15	nfrabw 2724	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}
17		csbeq1 3140	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
18		sbceq1a 3051	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ [𝑋 / 𝑥]𝜑))
19	17, 18	rabeqbidv 2807	. . . . . 6 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
20	12, 16, 19, 1	fvmptf 5769	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
21	8, 11, 20	syl2anc 411	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
22	21	eleq2d 2302	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}))
23		elrabi 2969	. . . . 5 ⊢ (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)
24	8, 23	anim12i 338	. . . 4 ⊢ ((𝑋 ∈ dom 𝐹 ∧ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
25	24	ex 115	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
26	22, 25	sylbid 150	. 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
27	6, 26	mpcom 36	1 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  {crab 2524  Vcvv 2812  [wsbc 3041  ⦋csb 3137   ↦ cmpt 4170  dom cdm 4748  Rel wrel 4753  Fun wfun 5345  ‘cfv 5351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fv 5359

This theorem is referenced by:  elfvmptrab  5772