Theorem elovmporab 7696

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 7691	. 2 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3	1	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ 𝑀 ∣ 𝜑}))
4		sbceq1a 3815	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
5		sbceq1a 3815	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
6	4, 5	sylan9bbr 510	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
7	6	adantl 481	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
8	7	rabbidv 3451	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → {𝑧 ∈ 𝑀 ∣ 𝜑} = {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
9		eqidd 2741	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
10		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
11		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
12		elovmporab.v	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)
13		rabexg 5355	. . . . . 6 ⊢ (𝑀 ∈ V → {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
14	12, 13	syl 17	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
15		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
16	15	nfel1 2925	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ V
17		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
18	17	nfel1 2925	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ V
19	16, 18	nfan 1898	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
20		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
21	20	nfel1 2925	. . . . . 6 ⊢ Ⅎ𝑦 𝑋 ∈ V
22		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑦𝑌
23	22	nfel1 2925	. . . . . 6 ⊢ Ⅎ𝑦 𝑌 ∈ V
24	21, 23	nfan 1898	. . . . 5 ⊢ Ⅎ𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
25		nfsbc1v 3824	. . . . . 6 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
26		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑥𝑀
27	25, 26	nfrabw 3483	. . . . 5 ⊢ Ⅎ𝑥{𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
28		nfsbc1v 3824	. . . . . . 7 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
29	20, 28	nfsbcw 3826	. . . . . 6 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
30		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑦𝑀
31	29, 30	nfrabw 3483	. . . . 5 ⊢ Ⅎ𝑦{𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
32	3, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31	ovmpodxf 7600	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
33	32	eleq2d 2830	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
34		df-3an 1089	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ 𝑀))
35	34	simplbi2com 502	. . . 4 ⊢ (𝑍 ∈ 𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
36		elrabi 3703	. . . 4 ⊢ (𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍 ∈ 𝑀)
37	35, 36	syl11 33	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
38	33, 37	sylbid 240	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
39	2, 38	mpcom 38	1 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488  [wsbc 3804  (class class class)co 7448   ∈ cmpo 7450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453

This theorem is referenced by: (None)