Theorem sbcfung 6525

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 3791	. . 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 5736	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcal 3803	. . . . 5 ⊢ ([𝐴 / 𝑥]∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤[𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦))
4		sbcex2 3804	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦[𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦))
5		sbcal 3803	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦))
6		sbcimg 3790	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑧 → [𝐴 / 𝑥]𝑧 = 𝑦)))
7		sbcbr123 5159	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧)
8		csbconstg 3874	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
9		csbconstg 3874	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
10	8, 9	breq12d 5118	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
11	7, 10	bitrid 282	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
12		sbcg 3818	. . . . . . . . . . . 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 278	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
15	14	albidv 1923	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
16	5, 15	bitrid 282	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
17	16	exbidv 1924	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦[𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
18	4, 17	bitrid 282	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
19	18	albidv 1923	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑤[𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
20	3, 19	bitrid 282	. . . 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 282	. 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 6510	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
24	23	sbcbii 3799	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
25		dffun3 6510	. 2 ⊢ (Fun ⦋𝐴 / 𝑥⦌𝐹 ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
26	22, 24, 25	3bitr4g 313	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539  ∃wex 1781   ∈ wcel 2106  [wsbc 3739  ⦋csb 3855   class class class wbr 5105  Rel wrel 5638  Fun wfun 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-fun 6498

This theorem is referenced by:  sbcfng  6665  esum2dlem  32691