Theorem sbcfung 5401

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 3094	. . 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 4861	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcal 3103	. . . . 5 ⊢ ([𝐴 / 𝑥]∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤[𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
4		sbcal 3103	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦[𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
5		sbcal 3103	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧))
6		sbcimg 3093	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧)))
7		sbcan 3094	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥](𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑦 ∧ [𝐴 / 𝑥]𝑤𝐹𝑧))
8		sbcbrg 4185	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦))
9		csbconstg 3161	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
10		csbconstg 3161	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
11	9, 10	breq12d 4143	. . . . . . . . . . . . . . 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 4185	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧))
14		csbconstg 3161	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
15	9, 14	breq12d 4143	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . 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 3121	. . . . . . . . . . . 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 1877	. . . . . . . . 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 1877	. . . . . . 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 1877	. . . . 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 477	. . 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 5387	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
31	30	sbcbii 3111	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∀𝑦∀𝑧((𝑤𝐹𝑦 ∧ 𝑤𝐹𝑧) → 𝑦 = 𝑧)))
32		dffun2 5387	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   ∈ wcel 2209  [wsbc 3051  ⦋csb 3147   class class class wbr 4130  Rel wrel 4779  Fun wfun 5371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-rel 4781  df-cnv 4782  df-co 4783  df-fun 5379

This theorem is used by:  sbcfng  5531