Theorem sbcfung 6602

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 3857	. . 3 ⊢ ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)) ↔ ([𝐴 / 𝑥]Rel 𝐹 ∧ [𝐴 / 𝑥]∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
2		sbcrel 5804	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcal 3868	. . . . 5 ⊢ ([𝐴 / 𝑥]∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤[𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦))
4		sbcex2 3869	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦[𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦))
5		sbcal 3868	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦))
6		sbcimg 3856	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑧 → [𝐴 / 𝑥]𝑧 = 𝑦)))
7		sbcbr123 5220	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧)
8		csbconstg 3940	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
9		csbconstg 3940	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
10	8, 9	breq12d 5179	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
11	7, 10	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
12		sbcg 3883	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑦 ↔ 𝑧 = 𝑦))
13	11, 12	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑧 → [𝐴 / 𝑥]𝑧 = 𝑦) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
14	6, 13	bitrd 279	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
15	14	albidv 1919	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
16	5, 15	bitrid 283	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
17	16	exbidv 1920	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦[𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
18	4, 17	bitrid 283	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
19	18	albidv 1919	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑤[𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
20	3, 19	bitrid 283	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
21	2, 20	anbi12d 631	. . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]Rel 𝐹 ∧ [𝐴 / 𝑥]∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)) ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦))))
22	1, 21	bitrid 283	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)) ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦))))
23		dffun3 6587	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
24	23	sbcbii 3865	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
25		dffun3 6587	. 2 ⊢ (Fun ⦋𝐴 / 𝑥⦌𝐹 ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
26	22, 24, 25	3bitr4g 314	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777   ∈ wcel 2108  [wsbc 3804  ⦋csb 3921   class class class wbr 5166  Rel wrel 5705  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6575

This theorem is referenced by:  sbcfng  6744  esum2dlem  34056