Theorem sbcfung 6568

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 5248 and ax-pr 5391. (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 3787	. . 3 ⊢ ([𝐴 / 𝑥](Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) ↔ ([𝐴 / 𝑥]Rel 𝐹 ∧ [𝐴 / 𝑥](𝐹 ∘ ◡𝐹) ⊆ I ))
2		sbcrel 5758	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel ⦋𝐴 / 𝑥⦌𝐹))
3		sbcssg 4476	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐹 ∘ ◡𝐹) ⊆ I ↔ ⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) ⊆ ⦋𝐴 / 𝑥⦌ I ))
4		csbcog 6303	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹 ∘ ◡𝐹) = (⦋𝐴 / 𝑥⦌𝐹 ∘ ⦋𝐴 / 𝑥⦌◡𝐹))
5		csbcnv 5865	. . . . . . . 8 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹
6	5	coeq2i 5839	. . . . . . 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 3942	. . . . 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 6545	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
14	13	sbcbii 3794	. 2 ⊢ ([𝐴 / 𝑥]Fun 𝐹 ↔ [𝐴 / 𝑥](Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
15		df-fun 6545	. 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 2144  [wsbc 3738  ⦋csb 3847   ⊆ wss 3896   I cid 5545  ^◡ccnv 5650   ∘ ccom 5655  Rel wrel 5656  Fun wfun 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6545

This theorem is used by:  sbcfng  6711  esum2dlem  34774