Theorem elovmporab1 7638

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. Usage of this theorem is discouraged because it depends on ax-13 2402. Use the weaker elovmporab1w 7637 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elovmporab1.o	⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
elovmporab1.v	⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)

Assertion

Ref	Expression
elovmporab1	⊢ (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   𝑧,𝑚

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 elovmporab1

Step	Hyp	Ref	Expression
1		elovmporab1.o	. . 3 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})
2	1	elmpocl 7631	. 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 3855	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
5	4	ad2antrl 738	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⦋𝑥 / 𝑚⦌𝑀 = ⦋𝑋 / 𝑚⦌𝑀)
6		sbceq1a 3755	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝜑 ↔ [𝑌 / 𝑦]𝜑))
7		sbceq1a 3755	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
8	6, 7	sylan9bbr 518	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
9	8	adantl 485	. . . . . 6 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
10	5, 9	rabeqbidv 3431	. . . . 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 2762	. . . . 5 ⊢ (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
12		simpl 486	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
13		simpr 488	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
14		elovmporab1.v	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)
15		rabexg 5292	. . . . . 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 2923	. . . . . . 7 ⊢ Ⅎ𝑥𝑋
18	17	nfel1 2939	. . . . . 6 ⊢ Ⅎ𝑥 𝑋 ∈ V
19		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥𝑌
20	19	nfel1 2939	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ V
21	18, 20	nfan 1918	. . . . 5 ⊢ Ⅎ𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑦𝑋
23	22	nfel1 2939	. . . . . 6 ⊢ Ⅎ𝑦 𝑋 ∈ V
24		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑦𝑌
25	24	nfel1 2939	. . . . . 6 ⊢ Ⅎ𝑦 𝑌 ∈ V
26	23, 25	nfan 1918	. . . . 5 ⊢ Ⅎ𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27		nfsbc1v 3764	. . . . . 6 ⊢ Ⅎ𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥𝑀
29	17, 28	nfcsb 3879	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑋 / 𝑚⦌𝑀
30	27, 29	nfrab 3451	. . . . 5 ⊢ Ⅎ𝑥{𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31		nfsbc1v 3764	. . . . . . 7 ⊢ Ⅎ𝑦[𝑌 / 𝑦]𝜑
32	22, 31	nfsbc 3769	. . . . . 6 ⊢ Ⅎ𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑦𝑀
34	22, 33	nfcsb 3879	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑋 / 𝑚⦌𝑀
35	32, 34	nfrab 3451	. . . . 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 7540	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
37	36	eleq2d 2847	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧 ∈ ⦋𝑋 / 𝑚⦌𝑀 ∣ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
38		df-3an 1099	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))
39	38	simplbi2com 506	. . . 4 ⊢ (𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀)))
40		elrabi 3646	. . . 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 242	. 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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453  [wsbc 3744  ⦋csb 3852  (class class class)co 7390   ∈ cmpo 7392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6471  df-fun 6517  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395

This theorem is referenced by: (None)