Theorem sbcfung 6591

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 5255 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 3791	. . 3 ⊢ ([𝐴 / 𝑥](Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) ↔ ([𝐴 / 𝑥]Rel 𝐹 ∧ [𝐴 / 𝑥](𝐹 ∘ ◡𝐹) ⊆ I ))
2		sbcrel 5765	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcssg 4480	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐹 ∘ ◡𝐹) ⊆ I ↔ ⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) ⊆ ⦋𝐴 / 𝑥⦌ I ))
4		csbcog 6326	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌◡𝐹))
5		csbcnv 5876	. . . . . . . 8 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹
6	5	coeq2i 5849	. . . . . . 7 ⊢ (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌◡𝐹)
7	4, 6	eqtr4di 2816	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ◡⦋𝐴 / 𝑥⦌𝐹))
8		csbconstg 3870	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ I = I )
9	7, 8	sseq12d 3946	. . . . 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 6568	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
14	13	sbcbii 3798	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
15		df-fun 6568	. 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 2147  [wsbc 3742  ⦋csb 3851   ⊆ wss 3900   I cid 5552  ^◡ccnv 5657   ∘ ccom 5662  Rel wrel 5663  Fun wfun 6560

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-rel 5665  df-cnv 5666  df-co 5667  df-fun 6568

This theorem is used by:  sbcfng  6736  esum2dlem  34835