Theorem csbfv12gALTVD 41240

Description: Virtual deduction proof of csbfv12 6715. 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 6715 is csbfv12gALTVD 41240 without virtual deductions and was automatically derived from csbfv12gALTVD 41240.

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 40915	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝐶   ▶   𝐴 ∈ 𝐶   )
2		sbceqg 4363	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐶 → ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦}))
3	1, 2	e1a 40968	. . . . . . . . . 10 ⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦})   )
4		csbima12 5949	. . . . . . . . . . . . . 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 40968	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵})   )
7		csbsng 4646	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵})
8	1, 7	e1a 40968	. . . . . . . . . . . . 13 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵}   )
9		imaeq2 5927	. . . . . . . . . . . . 13 ⊢ (⦋𝐴 / 𝑥⦌{𝐵} = {⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}))
10	8, 9	e1a 40968	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌𝐹 “ ⦋𝐴 / 𝑥⦌{𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   )
11		eqeq1 2827	. . . . . . . . . . . . 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 250	. . . . . . . . . . . 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 41029	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵})   )
14		csbconstg 3904	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌{𝑦} = {𝑦})
15	1, 14	e1a 40968	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦} = {𝑦}   )
16		eqeq12 2837	. . . . . . . . . . . 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 415	. . . . . . . . . . 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 41029	. . . . . . . . . 10 ⊢ (   𝐴 ∈ 𝐶   ▶   (⦋𝐴 / 𝑥⦌(𝐹 “ {𝐵}) = ⦋𝐴 / 𝑥⦌{𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
19		bibi1 354	. . . . . . . . . . 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 250	. . . . . . . . . 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 41029	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝐶   ▶   ([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
22	21	gen11 40957	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐶   ▶   ∀𝑦([𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦} ↔ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦})   )
23		abbi 2890	. . . . . . . . 9 ⊢ (∀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}})
24	23	biimpi 218	. . . . . . . 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 40968	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐶   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
26		csbab 4391	. . . . . . . . 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 40968	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ [𝐴 / 𝑥](𝐹 “ {𝐵}) = {𝑦}}   )
29		eqeq2 2835	. . . . . . . 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 231	. . . . . . 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 41029	. . . . . 6 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
32		unieq 4851	. . . . . 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 40968	. . . . 5 ⊢ (   𝐴 ∈ 𝐶   ▶   ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
34		csbuni 4869	. . . . . . 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 40968	. . . . 5 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
37		eqeq2 2835	. . . . . 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 231	. . . . 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 41029	. . . 4 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}} = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
40		dffv4 6669	. . . . . 6 ⊢ (𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
41	40	ax-gen 1796	. . . . 5 ⊢ ∀𝑥(𝐹‘𝐵) = ∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}
42		csbeq2 3890	. . . . . 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 41035	. . . 4 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌∪ {𝑦 ∣ (𝐹 “ {𝐵}) = {𝑦}}   )
45		eqeq2 2835	. . . . 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 231	. . . 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 41029	. . 3 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}   )
48		dffv4 6669	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = ∪ {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝐹 “ {⦋𝐴 / 𝑥⦌𝐵}) = {𝑦}}
49		eqeq2 2835	. . . 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 252	. . 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 41035	. 2 ⊢ (   𝐴 ∈ 𝐶   ▶   ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)   )
52	51	in1 40912	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537   ∈ wcel 2114  {cab 2801  [wsbc 3774  ⦋csb 3885  {csn 4569  ∪ cuni 4840   “ cima 5560  ‘cfv 6357

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fv 6365  df-vd1 40911

This theorem is referenced by: (None)