Theorem sbcfung 6545

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 3793	. . 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 5753	. . . 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 3792	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑧 → [𝐴 / 𝑥]𝑧 = 𝑦)))
7		sbcbr123 5154	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ ⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧)
8		csbconstg 3871	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
9		csbconstg 3871	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
10	8, 9	breq12d 5113	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑤⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
11	7, 10	bitrid 285	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧 ↔ 𝑤⦋𝐴 / 𝑥⦌𝐹𝑧))
12		sbcg 3816	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑦 ↔ 𝑧 = 𝑦))
13	11, 12	imbi12d 346	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑧 → [𝐴 / 𝑥]𝑧 = 𝑦) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
14	6, 13	bitrd 281	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ (𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
15	14	albidv 1940	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∀𝑧[𝐴 / 𝑥](𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
16	5, 15	bitrid 285	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
17	16	exbidv 1941	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦[𝐴 / 𝑥]∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
18	4, 17	bitrid 285	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
19	18	albidv 1940	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑤[𝐴 / 𝑥]∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
20	3, 19	bitrid 285	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
21	2, 20	anbi12d 641	. . 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 285	. 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 6533	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
24	23	sbcbii 3800	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤𝐹𝑧 → 𝑧 = 𝑦)))
25		dffun3 6533	. 2 ⊢ (Fun ⦋𝐴 / 𝑥⦌𝐹 ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ ∀𝑤∃𝑦∀𝑧(𝑤⦋𝐴 / 𝑥⦌𝐹𝑧 → 𝑧 = 𝑦)))
26	22, 24, 25	3bitr4g 316	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558  ∃wex 1799   ∈ wcel 2142  [wsbc 3744  ⦋csb 3852   class class class wbr 5100  Rel wrel 5652  Fun wfun 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-fun 6523

This theorem is referenced by:  sbcfng  6688  esum2dlem  34389