Theorem elfvmptrab1w 6786

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. Version of elfvmptrab1 6787 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.)

Hypotheses

Ref	Expression
elfvmptrab1w.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
elfvmptrab1w.v	⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)

Assertion

Ref	Expression
elfvmptrab1w	⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))

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

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

Proof of Theorem elfvmptrab1w

Step	Hyp	Ref	Expression
1		elfvdm 6691	. 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → 𝑋 ∈ dom 𝐹)
2		elfvmptrab1w.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
3	2	dmmptss 6071	. . . . . 6 ⊢ dom 𝐹 ⊆ 𝑉
4	3	sseli 3889	. . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉)
5		elfvmptrab1w.v	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
6		rabexg 5202	. . . . . 6 ⊢ (⦋𝑋 / 𝑚⦌𝑀 ∈ V → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V)
7	4, 5, 6	3syl 18	. . . . 5 ⊢ (𝑋 ∈ dom 𝐹 → {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V)
8		nfcv 2920	. . . . . 6 ⊢ Ⅎ𝑥𝑋
9		nfsbc1v 3717	. . . . . . 7 ⊢ Ⅎ𝑥[𝑋 / 𝑥]𝜑
10		nfcv 2920	. . . . . . . 8 ⊢ Ⅎ𝑥𝑀
11	8, 10	nfcsbw 3832	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀
12	9, 11	nfrabw 3304	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}
13		csbeq1 3809	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
14		sbceq1a 3708	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ [𝑋 / 𝑥]𝜑))
15	13, 14	rabeqbidv 3399	. . . . . 6 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
16	8, 12, 15, 2	fvmptf 6781	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} ∈ V) → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
17	4, 7, 16	syl2anc 588	. . . 4 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑})
18	17	eleq2d 2838	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}))
19		elrabi 3597	. . . . 5 ⊢ (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)
20	4, 19	anim12i 616	. . . 4 ⊢ ((𝑋 ∈ dom 𝐹 ∧ 𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑}) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
21	20	ex 417	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥]𝜑} → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
22	18, 21	sylbid 243	. 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
23	1, 22	mpcom 38	1 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {crab 3075  Vcvv 3410  [wsbc 3697  ⦋csb 3806   ↦ cmpt 5113  dom cdm 5525  ‘cfv 6336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fv 6344

This theorem is referenced by:  elfvmptrab  6788