Theorem sbcfung 4925

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 2805	. . 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 4426	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcal 2810	. . . . 5 ⊢ ([𝐴 / 𝑥]∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤[𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
4		sbcal 2810	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦[𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
5		sbcal 2810	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
6		sbcimg 2804	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧)))
7		sbcan 2805	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑦 ∧ [𝐴 / 𝑥]𝑤𝐹𝑧))
8		sbcbrg 3813	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦))
9		csbconstg 2864	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
10		csbconstg 2864	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
11	9, 10	breq12d 3777	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑦))
12	8, 11	bitrd 177	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑦))
13		sbcbrg 3813	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧))
14		csbconstg 2864	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
15	9, 14	breq12d 3777	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
16	13, 15	bitrd 177	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
17	12, 16	anbi12d 442	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑦 ∧ [𝐴 / 𝑥]𝑤𝐹𝑧) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧)))
18	7, 17	syl5bb 181	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧)))
19		sbcg 2827	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 = 𝑧 ↔ 𝑦 = 𝑧))
20	18, 19	imbi12d 223	. . . . . . . . . . 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 177	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
22	21	albidv 1705	. . . . . . . . 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 181	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤⦋𝐴 / 𝑥⦌𝐹𝑦 ∧ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧) → 𝑦 = 𝑧)))
24	23	albidv 1705	. . . . . . 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 181	. . . . . 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 1705	. . . . 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 181	. . . 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 442	. . 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 181	. 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 4912	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
31	30	sbcbii 2818	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
32		dffun2 4912	. 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 212	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   ∈ wcel 1393  [wsbc 2764  ⦋csb 2852   class class class wbr 3764  Rel wrel 4350  Fun wfun 4896

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-rel 4352  df-cnv 4353  df-co 4354  df-fun 4904

This theorem is referenced by:  sbcfng  5044