Theorem elovmpt2rab 6778

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
elovmpt2rab.o	⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ 𝑀 ∣ 𝜑})
elovmpt2rab.v	⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)

Assertion

Ref	Expression
elovmpt2rab	⊢ (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 elovmpt2rab

Step	Hyp	Ref	Expression
1		elovmpt2rab.o	. . 3 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ 𝑀 ∣ 𝜑})
2	1	elmpt2cl 6774	. 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 3413	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
5		sbceq1a 3413	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
6	4, 5	sylan9bbr 733	. . . . . . 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 3164	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → {𝑧 ∈ 𝑀 ∣ 𝜑} = {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
9		eqidd 2611	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
10		simpl 472	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
11		simpr 476	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
12		elovmpt2rab.v	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)
13		rabexg 4739	. . . . . 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 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
16	15	nfel1 2765	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ V
17		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
18	17	nfel1 2765	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ V
19	16, 18	nfan 1816	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
20		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
21	20	nfel1 2765	. . . . . 6 ⊢ Ⅎ𝑦 𝑋 ∈ V
22		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑦𝑌
23	22	nfel1 2765	. . . . . 6 ⊢ Ⅎ𝑦 𝑌 ∈ V
24	21, 23	nfan 1816	. . . . 5 ⊢ Ⅎ𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
25		nfsbc1v 3422	. . . . . 6 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
26		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝑀
27	25, 26	nfrab 3100	. . . . 5 ⊢ Ⅎ𝑥{𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
28		nfsbc1v 3422	. . . . . . 7 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
29	20, 28	nfsbc 3424	. . . . . 6 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
30		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑦𝑀
31	29, 30	nfrab 3100	. . . . 5 ⊢ Ⅎ𝑦{𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
32	3, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31	ovmpt2dxf 6684	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
33	32	eleq2d 2673	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
34		elrabi 3328	. . . . 5 ⊢ (𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍 ∈ 𝑀)
35		df-3an 1033	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ 𝑀))
36	35	simplbi2com 655	. . . . 5 ⊢ (𝑍 ∈ 𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
37	34, 36	syl 17	. . . 4 ⊢ (𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
38	37	com12 32	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ {𝑧 ∈ 𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
39	33, 38	sylbid 229	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀)))
40	2, 39	mpcom 37	1 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ 𝑀))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  [wsbc 3402  (class class class)co 6549   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by: (None)