Theorem sbcfung 6557

Description: Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) Shorten proof and remove dependency on ax-sep 5251 and ax-pr 5398. (Revised by Eric Schmidt, 12-Sep-2026.)

Assertion

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

Proof of Theorem sbcfung

Step	Hyp	Ref	Expression
1		sbcan 3788	. . 3 ⊢ ([𝐴 / 𝑥](Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) ↔ ([𝐴 / 𝑥]Rel 𝐹 ∧ [𝐴 / 𝑥](𝐹 ∘ ◡𝐹) ⊆ I ))
2		sbcrel 5761	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcssg 4477	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐹 ∘ ◡𝐹) ⊆ I ↔ ⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) ⊆ ⦋𝐴 / 𝑥⦌ I ))
4		csbcog 6295	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌◡𝐹))
5		csbcnv 5866	. . . . . . . 8 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹
6	5	coeq2i 5840	. . . . . . 7 ⊢ (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌◡𝐹)
7	4, 6	eqtr4di 2813	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹))
8		csbconstg 3866	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ I = I )
9	7, 8	sseq12d 3964	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) ⊆ ⦋𝐴 / 𝑥⦌ I ↔ (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹) ⊆ I ))
10	3, 9	bitrd 282	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐹 ∘ ◡𝐹) ⊆ I ↔ (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹) ⊆ I ))
11	2, 10	anbi12d 644	. . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]Rel 𝐹 ∧ [𝐴 / 𝑥](𝐹 ∘ ◡𝐹) ⊆ I ) ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹) ⊆ I )))
12	1, 11	bitrid 286	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹) ⊆ I )))
13		df-fun 6535	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
14	13	sbcbii 3795	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
15		df-fun 6535	. 2 ⊢ (Fun ⦋𝐴 / 𝑥⦌𝐹 ↔ (Rel ⦋𝐴 / 𝑥⦌𝐹 ∧ (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹) ⊆ I ))
16	12, 14, 15	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   ∈ wcel 2145  [wsbc 3739  ⦋csb 3847   ⊆ wss 3899   I cid 5549  ^◡ccnv 5654   ∘ ccom 5659  Rel wrel 5660  Fun wfun 6527

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 2732

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-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  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-rel 5662  df-cnv 5663  df-co 5664  df-fun 6535

This theorem is used by:  sbcfng  6700  esum2dlem  34603