Theorem elovmporab1w 7670

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 7671 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by Alexander van der Vekens, 15-Jul-2018.) Avoid ax-13 2407. (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 7664	. 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 3859	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
5	4	ad2antrl 741	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
6		sbceq1a 3758	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
7		sbceq1a 3758	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
8	6, 7	sylan9bbr 520	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
9	8	adantl 487	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
10	5, 9	rabeqbidv 3437	. . . . 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 2767	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
12		simpl 488	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
13		simpr 490	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
14		elovmporab1w.v	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
15		rabexg 5313	. . . . . 6 ⊢ (⦋𝑋 / 𝑚⦌𝑀 ∈ V → {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
16	14, 15	syl 18	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17		nfcv 2928	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
18	17	nfel1 2944	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ V
19		nfcv 2928	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
20	19	nfel1 2944	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ V
21	18, 20	nfan 1932	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22		nfcv 2928	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
23	22	nfel1 2944	. . . . . 6 ⊢ Ⅎ𝑦 𝑋 ∈ V
24		nfcv 2928	. . . . . . 7 ⊢ Ⅎ𝑦𝑌
25	24	nfel1 2944	. . . . . 6 ⊢ Ⅎ𝑦 𝑌 ∈ V
26	23, 25	nfan 1932	. . . . 5 ⊢ Ⅎ𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27		nfsbc1v 3767	. . . . . 6 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28		nfcv 2928	. . . . . . 7 ⊢ Ⅎ𝑥𝑀
29	17, 28	nfcsbw 3882	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀
30	27, 29	nfrabw 3455	. . . . 5 ⊢ Ⅎ𝑥{𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31		nfsbc1v 3767	. . . . . . 7 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
32	22, 31	nfsbcw 3769	. . . . . 6 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33		nfcv 2928	. . . . . . 7 ⊢ Ⅎ𝑦𝑀
34	22, 33	nfcsbw 3882	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑋 / 𝑚⦌𝑀
35	32, 34	nfrabw 3455	. . . . 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 7573	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
37	36	eleq2d 2852	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
38		df-3an 1105	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))
39	38	simplbi2com 508	. . . 4 ⊢ (𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
40		elrabi 3649	. . . 4 ⊢ (𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)
41	39, 40	syl11 34	. . 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 243	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
43	2, 42	mpcom 39	1 ⊢ (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {crab 3419  Vcvv 3458  [wsbc 3747  ⦋csb 3856  (class class class)co 7423   ∈ cmpo 7425

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428

This theorem is used by:  elovmpowrd  14615