Theorem elovmporab1w 6290

Description: Implications for the value of an operation, 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 first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by GG, 26-Jan-2024.)

Hypotheses

Ref	Expression
elovmporab1w.o	⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
elovmporab1w.v	⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)

Assertion

Ref	Expression
elovmporab1w	⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))

Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑧,𝑍   𝑥,𝑚,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑚)   𝑀(𝑚)   𝑂(𝑥, 𝑦, 𝑧, 𝑚)   𝑋(𝑚)   𝑌(𝑚)   𝑍(𝑥, 𝑦, 𝑚)

Proof of Theorem elovmporab1w

Step	Hyp	Ref	Expression
1		elovmporab1w.o	. . 3 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
2	1	elmpocl 6284	. 2 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3	1	a1i 9	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}))
4		csbeq1 3150	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
5	4	ad2antrl 494	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
6		sbceq1a 3061	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
7		sbceq1a 3061	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
8	6, 7	sylan9bbr 467	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
9	8	adantl 277	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
10	5, 9	rabeqbidv 2816	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑} = {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
11		eqidd 2239	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
12		simpl 109	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
13		simpr 110	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
14		elovmporab1w.v	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
15		rabexg 4279	. . . . . 6 ⊢ (⦋𝑋 / 𝑚⦌𝑀 ∈ V → {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
16	14, 15	syl 14	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
18	17	nfel1 2403	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ V
19		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
20	19	nfel1 2403	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ V
21	18, 20	nfan 1618	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
23	22	nfel1 2403	. . . . . 6 ⊢ Ⅎ𝑦 𝑋 ∈ V
24		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑦𝑌
25	24	nfel1 2403	. . . . . 6 ⊢ Ⅎ𝑦 𝑌 ∈ V
26	23, 25	nfan 1618	. . . . 5 ⊢ Ⅎ𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27		nfsbc1v 3070	. . . . . 6 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑥𝑀
29	17, 28	nfcsbw 3184	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀
30	27, 29	nfrabw 2733	. . . . 5 ⊢ Ⅎ𝑥{𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31		nfsbc1v 3070	. . . . . . 7 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
32	22, 31	nfsbcw 3182	. . . . . 6 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑦𝑀
34	22, 33	nfcsbw 3184	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑋 / 𝑚⦌𝑀
35	32, 34	nfrabw 2733	. . . . 5 ⊢ Ⅎ𝑦{𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
36	3, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35	ovmpodxf 6214	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
37	36	eleq2d 2308	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
38		df-3an 1011	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))
39	38	simplbi2com 1494	. . . 4 ⊢ (𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
40		elrabi 2979	. . . 4 ⊢ (𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)
41	39, 40	syl11 31	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
42	37, 41	sylbid 150	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
43	2, 42	mpcom 36	1 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821  [wsbc 3051  ⦋csb 3147  (class class class)co 6085   ∈ cmpo 6087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090

This theorem is used by:  elovmpowrd  11362