Theorem elovmporab1w 7663

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. Version of elovmporab1 7664 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Alexander van der Vekens, 15-Jul-2018.) Avoid ax-13 2375. (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 7657	. 2 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
3	1	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}))
4		csbeq1 3884	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
5	4	ad2antrl 728	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
6		sbceq1a 3783	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
7		sbceq1a 3783	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
8	6, 7	sylan9bbr 510	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
9	8	adantl 481	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
10	5, 9	rabeqbidv 3439	. . . . 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 2735	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
12		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
13		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
14		elovmporab1w.v	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
15		rabexg 5319	. . . . . 6 ⊢ (⦋𝑋 / 𝑚⦌𝑀 ∈ V → {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
16	14, 15	syl 17	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
18	17	nfel1 2914	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ V
19		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
20	19	nfel1 2914	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ V
21	18, 20	nfan 1898	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
23	22	nfel1 2914	. . . . . 6 ⊢ Ⅎ𝑦 𝑋 ∈ V
24		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑦𝑌
25	24	nfel1 2914	. . . . . 6 ⊢ Ⅎ𝑦 𝑌 ∈ V
26	23, 25	nfan 1898	. . . . 5 ⊢ Ⅎ𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27		nfsbc1v 3792	. . . . . 6 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝑀
29	17, 28	nfcsbw 3907	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀
30	27, 29	nfrabw 3459	. . . . 5 ⊢ Ⅎ𝑥{𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31		nfsbc1v 3792	. . . . . . 7 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
32	22, 31	nfsbcw 3794	. . . . . 6 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑦𝑀
34	22, 33	nfcsbw 3907	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑋 / 𝑚⦌𝑀
35	32, 34	nfrabw 3459	. . . . 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 7566	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
37	36	eleq2d 2819	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
38		df-3an 1088	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))
39	38	simplbi2com 502	. . . 4 ⊢ (𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
40		elrabi 3671	. . . 4 ⊢ (𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)
41	39, 40	syl11 33	. . 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 240	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
43	2, 42	mpcom 38	1 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  {crab 3420  Vcvv 3464  [wsbc 3772  ⦋csb 3881  (class class class)co 7414   ∈ cmpo 7416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6495  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419

This theorem is referenced by:  elovmpowrd  14579