Theorem sbcfungOLD 6564

Description: Obsolete version of sbcfung 6563 as of 12-Sep-2026. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcfungOLD	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

Proof of Theorem sbcfungOLD

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcan 3788	. . 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 5757	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcal 3798	. . . . 5 ⊢ ([𝐴 / 𝑥]∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤[𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦))
4		sbcex2 3799	. . . . . . 7 ⊢ ([𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦[𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦))
5		sbcal 3798	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦))
6		sbcimg 3787	. . . . . . . . . . 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 3866	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
9		csbconstg 3866	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
10	8, 9	breq12d 5116	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
11	7, 10	bitrid 286	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
12		sbcg 3811	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑦 ↔ 𝑧 = 𝑦))
13	11, 12	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑧 → [𝐴 / 𝑥]𝑧 = 𝑦) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
14	6, 13	bitrd 282	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
15	14	albidv 1953	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
16	5, 15	bitrid 286	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
17	16	exbidv 1954	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦[𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
18	4, 17	bitrid 286	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
19	18	albidv 1953	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑤[𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
20	3, 19	bitrid 286	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
21	2, 20	anbi12d 644	. . 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 286	. 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 6550	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
24	23	sbcbii 3795	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
25		dffun3 6550	. 2 ⊢ (Fun ⦋𝐴 / 𝑥⦌𝐹 ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
26	22, 24, 25	3bitr4g 317	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812   ∈ wcel 2145  [wsbc 3739  ⦋csb 3847   class class class wbr 5103  Rel wrel 5656  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6540

This theorem is used by: (None)