Theorem sbcfung 5212

Description: Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Assertion

Ref	Expression
sbcfung	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

Proof of Theorem sbcfung

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcan 2993	. . 3 ⊢ ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ ([𝐴 / 𝑥]Rel 𝐹 ∧ [𝐴 / 𝑥]∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
2		sbcrel 4690	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcal 3002	. . . . 5 ⊢ ([𝐴 / 𝑥]∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤[𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
4		sbcal 3002	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦[𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
5		sbcal 3002	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
6		sbcimg 2992	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧)))
7		sbcan 2993	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑦 ∧ [𝐴 / 𝑥]𝑤𝐹𝑧))
8		sbcbrg 4036	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦))
9		csbconstg 3059	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
10		csbconstg 3059	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
11	9, 10	breq12d 3995	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑦))
12	8, 11	bitrd 187	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑦))
13		sbcbrg 4036	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧))
14		csbconstg 3059	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
15	9, 14	breq12d 3995	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
16	13, 15	bitrd 187	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
17	12, 16	anbi12d 465	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑦 ∧ [𝐴 / 𝑥]𝑤𝐹𝑧) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧)))
18	7, 17	syl5bb 191	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧)))
19		sbcg 3020	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
20	18, 19	imbi12d 233	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧) ↔ ((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
21	6, 20	bitrd 187	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
22	21	albidv 1812	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
23	5, 22	syl5bb 191	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
24	23	albidv 1812	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦[𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
25	4, 24	syl5bb 191	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
26	25	albidv 1812	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑤[𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤∀𝑦∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
27	3, 26	syl5bb 191	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤∀𝑦∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
28	2, 27	anbi12d 465	. . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]Rel 𝐹 ∧ [𝐴 / 𝑥]∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧))))
29	1, 28	syl5bb 191	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧))))
30		dffun2 5198	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
31	30	sbcbii 3010	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
32		dffun2 5198	. 2 ⊢ (Fun ⦋𝐴 / 𝑥⦌𝐹 ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
33	29, 31, 32	3bitr4g 222	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   ∈ wcel 2136  [wsbc 2951  ⦋csb 3045   class class class wbr 3982  Rel wrel 4609  Fun wfun 5182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-rel 4611  df-cnv 4612  df-co 4613  df-fun 5190

This theorem is referenced by:  sbcfng  5335