Theorem sbcfg 5383

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 5239	. . . 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 3036	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ [𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)))
4		sbcfng 5382	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
5		sbcssg 3547	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
6		csbrng 5108	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌ran 𝐹 = ran ⦋𝑋 / 𝑥⦌𝐹)
7	6	sseq1d 3199	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
8	5, 7	bitrd 188	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
9	4, 8	anbi12d 473	. . 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 3020	. . 3 ⊢ ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴 ∧ [𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵))
11		df-f 5239	. . 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 223	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))
13	3, 12	bitrd 188	1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2160  [wsbc 2977  ⦋csb 3072   ⊆ wss 3144  ran crn 4645   Fn wfn 5230  ⟶wf 5231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239

This theorem is referenced by:  ctiunctlemf  12492