Theorem sbcfg 5336

Description: Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Assertion

Ref	Expression
sbcfg	⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))

Distinct variable groups:   𝑥,𝑉   𝑥,𝑋

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem sbcfg

Step	Hyp	Ref	Expression
1		df-f 5192	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2	1	a1i 9	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)))
3	2	sbcbidv 3009	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ [𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)))
4		sbcfng 5335	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
5		sbcssg 3518	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
6		csbrng 5065	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌ran 𝐹 = ran ⦋𝑋 / 𝑥⦌𝐹)
7	6	sseq1d 3171	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
8	5, 7	bitrd 187	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
9	4, 8	anbi12d 465	. . 3 ⊢ (𝑋 ∈ 𝑉 → (([𝑋 / 𝑥]𝐹 Fn 𝐴 ∧ [𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵) ↔ (⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴 ∧ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵)))
10		sbcan 2993	. . 3 ⊢ ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴 ∧ [𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵))
11		df-f 5192	. . 3 ⊢ (⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵 ↔ (⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴 ∧ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
12	9, 10, 11	3bitr4g 222	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))
13	3, 12	bitrd 187	1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2136  [wsbc 2951  ⦋csb 3045   ⊆ wss 3116  ran crn 4605   Fn wfn 5183  ⟶wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  ctiunctlemf  12371