Theorem csbfv12gALTVD 45139

Description: Virtual deduction proof of csbfv12 6879. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbfv12 6879 is csbfv12gALTVD 45139 without virtual deductions and was automatically derived from csbfv12gALTVD 45139.

1::	⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   )
2:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦} = { 𝑦}   )
3:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵 }) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵})   )
4:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { ⦋𝐴 / 𝑥⦌𝐵}   )
5:4:	⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   )
6:3,5:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵 }) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   )
7:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦})   )
8:6,2:	⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌(𝐹 “ { 𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
9:7,8:	⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ { 𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})    )
10:9:	⊢ (   𝐴 ∈ 𝐶   ▶   ∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
11:10:	⊢ (   𝐴 ∈ 𝐶   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
12:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   )
13:11,12:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦 }}   )
14:13:	⊢ (   𝐴 ∈ 𝐶   ▶   ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
15:1:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
16:14,15:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ ( 𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
17::	⊢ (𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
18:17:	⊢ ∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵 }) = {𝑦}}
19:1,18:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
20:16,19:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
21::	⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}
22:20,21:	⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)   )
qed:22:	⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbfv12gALTVD	⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))

Proof of Theorem csbfv12gALTVD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn1 44815	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   )
2		sbceqg 4364	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐶 → ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦}))
3	1, 2	e1a 44868	. . . . . . . . . 10 ⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦})   )
4		csbima12 6038	. . . . . . . . . . . . . 14 ⊢ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵})
5	4	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}))
6	1, 5	e1a 44868	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵})   )
7		csbsng 4665	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})
8	1, 7	e1a 44868	. . . . . . . . . . . . 13 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}   )
9		imaeq2 6015	. . . . . . . . . . . . 13 ⊢ (⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}))
10	8, 9	e1a 44868	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   )
11		eqeq1 2740	. . . . . . . . . . . . 13 ⊢ (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) → (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) ↔ (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})))
12	11	biimprd 248	. . . . . . . . . . . 12 ⊢ (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) → ((⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) → ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})))
13	6, 10, 12	e11 44929	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   )
14		csbconstg 3868	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌{𝑦} = {𝑦})
15	1, 14	e1a 44868	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦} = {𝑦}   )
16		eqeq12 2753	. . . . . . . . . . . 12 ⊢ ((⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) ∧ ⦋𝐴 / 𝑥⦌{𝑦} = {𝑦}) → (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}))
17	16	ex 412	. . . . . . . . . . 11 ⊢ (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) → (⦋𝐴 / 𝑥⦌{𝑦} = {𝑦} → (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})))
18	13, 15, 17	e11 44929	. . . . . . . . . 10 ⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
19		bibi1 351	. . . . . . . . . . 11 ⊢ (([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦}) → (([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}) ↔ (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})))
20	19	biimprd 248	. . . . . . . . . 10 ⊢ (([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦}) → ((⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}) → ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})))
21	3, 18, 20	e11 44929	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
22	21	gen11 44857	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐶   ▶   ∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
23		abbib 2805	. . . . . . . . 9 ⊢ ({𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} ↔ ∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}))
24	23	biimpri 228	. . . . . . . 8 ⊢ (∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}) → {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}})
25	22, 24	e1a 44868	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐶   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
26		csbab 4392	. . . . . . . . 9 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}
27	26	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}})
28	1, 27	e1a 44868	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   )
29		eqeq2 2748	. . . . . . . 8 ⊢ ({𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → (⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} ↔ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}))
30	29	biimpd 229	. . . . . . 7 ⊢ ({𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → (⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} → ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}))
31	25, 28, 30	e11 44929	. . . . . 6 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
32		unieq 4874	. . . . . 6 ⊢ (⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}})
33	31, 32	e1a 44868	. . . . 5 ⊢ (   𝐴 ∈ 𝐶   ▶   ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
34		csbuni 4893	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
35	34	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}})
36	1, 35	e1a 44868	. . . . 5 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
37		eqeq2 2748	. . . . . 6 ⊢ (∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → (⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} ↔ ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}))
38	37	biimpd 229	. . . . 5 ⊢ (∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → (⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} → ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}))
39	33, 36, 38	e11 44929	. . . 4 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
40		dffv4 6831	. . . . . 6 ⊢ (𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
41	40	ax-gen 1796	. . . . 5 ⊢ ∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
42		csbeq2 3854	. . . . . 6 ⊢ (∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}})
43	42	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}))
44	1, 41, 43	e10 44935	. . . 4 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
45		eqeq2 2748	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → (⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} ↔ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}))
46	45	biimpd 229	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → (⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}))
47	39, 44, 46	e11 44929	. . 3 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
48		dffv4 6831	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}
49		eqeq2 2748	. . . 4 ⊢ ((⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → (⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) ↔ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}))
50	49	biimprcd 250	. . 3 ⊢ (⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → ((⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}} → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)))
51	47, 48, 50	e10 44935	. 2 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)   )
52	51	in1 44812	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {cab 2714  [wsbc 3740  ⦋csb 3849  {csn 4580  ∪ cuni 4863   “ cima 5627  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fv 6500  df-vd1 44811

This theorem is referenced by: (None)