Theorem elovmporab1w 6154

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 6148	. 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 3097	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
5	4	ad2antrl 490	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
6		sbceq1a 3009	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
7		sbceq1a 3009	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
8	6, 7	sylan9bbr 463	. . . . . . 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 2768	. . . . 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 2207	. . . . 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 4191	. . . . . 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 2349	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
18	17	nfel1 2360	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ V
19		nfcv 2349	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
20	19	nfel1 2360	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ V
21	18, 20	nfan 1589	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22		nfcv 2349	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
23	22	nfel1 2360	. . . . . 6 ⊢ Ⅎ𝑦 𝑋 ∈ V
24		nfcv 2349	. . . . . . 7 ⊢ Ⅎ𝑦𝑌
25	24	nfel1 2360	. . . . . 6 ⊢ Ⅎ𝑦 𝑌 ∈ V
26	23, 25	nfan 1589	. . . . 5 ⊢ Ⅎ𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27		nfsbc1v 3018	. . . . . 6 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28		nfcv 2349	. . . . . . 7 ⊢ Ⅎ𝑥𝑀
29	17, 28	nfcsbw 3131	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀
30	27, 29	nfrabw 2688	. . . . 5 ⊢ Ⅎ𝑥{𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31		nfsbc1v 3018	. . . . . . 7 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
32	22, 31	nfsbcw 3129	. . . . . 6 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33		nfcv 2349	. . . . . . 7 ⊢ Ⅎ𝑦𝑀
34	22, 33	nfcsbw 3131	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑋 / 𝑚⦌𝑀
35	32, 34	nfrabw 2688	. . . . 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 6078	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
37	36	eleq2d 2276	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
38		df-3an 983	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))
39	38	simplbi2com 1465	. . . 4 ⊢ (𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
40		elrabi 2927	. . . 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))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  {crab 2489  Vcvv 2773  [wsbc 2999  ⦋csb 3094  (class class class)co 5951   ∈ cmpo 5953

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-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-setind 4589

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-iota 5237  df-fun 5278  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956

This theorem is referenced by:  elovmpowrd  11042