Theorem elovmporab 6227

Description: Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Hypotheses

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

Assertion

Ref	Expression
elovmporab	⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀))

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝑂(𝑥,𝑦,𝑧)   𝑍(𝑥,𝑦)

Proof of Theorem elovmporab

Step	Hyp	Ref	Expression
1		elovmporab.o	. . 3 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ 𝑀 ∣ 𝜑})
2	1	elmpocl 6222	. 2 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3	1	a1i 9	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ 𝑀 ∣ 𝜑}))
4		sbceq1a 3040	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
5		sbceq1a 3040	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
6	4, 5	sylan9bbr 463	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
7	6	adantl 277	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
8	7	rabbidv 2790	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → {𝑧 ∈ 𝑀 ∣ 𝜑} = {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
9		eqidd 2231	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
10		simpl 109	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
11		simpr 110	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
12		elovmporab.v	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)
13		rabexg 4234	. . . . . 6 ⊢ (𝑀 ∈ V → {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
14	12, 13	syl 14	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
15		nfcv 2373	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
16	15	nfel1 2384	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ V
17		nfcv 2373	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
18	17	nfel1 2384	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ V
19	16, 18	nfan 1613	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
20		nfcv 2373	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
21	20	nfel1 2384	. . . . . 6 ⊢ Ⅎ𝑦 𝑋 ∈ V
22		nfcv 2373	. . . . . . 7 ⊢ Ⅎ𝑦𝑌
23	22	nfel1 2384	. . . . . 6 ⊢ Ⅎ𝑦 𝑌 ∈ V
24	21, 23	nfan 1613	. . . . 5 ⊢ Ⅎ𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
25		nfsbc1v 3049	. . . . . 6 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
26		nfcv 2373	. . . . . 6 ⊢ Ⅎ𝑥𝑀
27	25, 26	nfrabw 2713	. . . . 5 ⊢ Ⅎ𝑥{𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
28		nfsbc1v 3049	. . . . . . 7 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
29	20, 28	nfsbcw 3161	. . . . . 6 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
30		nfcv 2373	. . . . . 6 ⊢ Ⅎ𝑦𝑀
31	29, 30	nfrabw 2713	. . . . 5 ⊢ Ⅎ𝑦{𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
32	3, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31	ovmpodxf 6152	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
33	32	eleq2d 2300	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
34		df-3an 1006	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ 𝑀))
35	34	simplbi2com 1489	. . . 4 ⊢ (𝑍 ∈ 𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
36		elrabi 2958	. . . 4 ⊢ (𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍 ∈ 𝑀)
37	35, 36	syl11 31	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
38	33, 37	sylbid 150	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
39	2, 38	mpcom 36	1 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2201  {crab 2513  Vcvv 2801  [wsbc 3030  (class class class)co 6023   ∈ cmpo 6025

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-setind 4637

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-iota 5288  df-fun 5330  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028

This theorem is referenced by: (None)