Theorem sbcfung 5278

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 3028	. . 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 4745	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcal 3037	. . . . 5 ⊢ ([𝐴 / 𝑥]∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤[𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
4		sbcal 3037	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦[𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
5		sbcal 3037	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
6		sbcimg 3027	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧)))
7		sbcan 3028	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑦 ∧ [𝐴 / 𝑥]𝑤𝐹𝑧))
8		sbcbrg 4083	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦))
9		csbconstg 3094	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
10		csbconstg 3094	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
11	9, 10	breq12d 4042	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑦))
12	8, 11	bitrd 188	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑦))
13		sbcbrg 4083	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧))
14		csbconstg 3094	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
15	9, 14	breq12d 4042	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
16	13, 15	bitrd 188	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
17	12, 16	anbi12d 473	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑦 ∧ [𝐴 / 𝑥]𝑤𝐹𝑧) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧)))
18	7, 17	bitrid 192	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧)))
19		sbcg 3055	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
20	18, 19	imbi12d 234	. . . . . . . . . . 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 188	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
22	21	albidv 1835	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
23	5, 22	bitrid 192	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
24	23	albidv 1835	. . . . . . 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	bitrid 192	. . . . . 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 1835	. . . . 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	bitrid 192	. . . 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 473	. . 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	bitrid 192	. 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 5264	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
31	30	sbcbii 3045	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
32		dffun2 5264	. 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 223	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   ∈ wcel 2164  [wsbc 2985  ⦋csb 3080   class class class wbr 4029  Rel wrel 4664  Fun wfun 5248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-rel 4666  df-cnv 4667  df-co 4668  df-fun 5256

This theorem is referenced by:  sbcfng  5401