Theorem sbcfg 6689

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 6525	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2	1	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)))
3	2	sbcbidv 3799	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ [𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)))
4		sbcfng 6688	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
5		sbcssg 4475	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
6		csbrn 6190	. . . . . 6 ⊢ ⦋𝑋 / 𝑥⦌ran 𝐹 = ran ⦋𝑋 / 𝑥⦌𝐹
7	6	sseq1i 3964	. . . . 5 ⊢ (⦋𝑋 / 𝑥⦌ran 𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵)
8	5, 7	bitrdi 289	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵 ↔ ran ⦋𝑋 / 𝑥⦌𝐹 ⊆ ⦋𝑋 / 𝑥⦌𝐵))
9	4, 8	anbi12d 641	. . 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 3793	. . 3 ⊢ ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴 ∧ [𝑋 / 𝑥]ran 𝐹 ⊆ 𝐵))
11		df-f 6525	. . 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 316	. 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))
13	3, 12	bitrd 281	1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2142  [wsbc 3744  ⦋csb 3852   ⊆ wss 3904  ran crn 5648   Fn wfn 6516  ⟶wf 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-fun 6523  df-fn 6524  df-f 6525

This theorem is referenced by:  csbwrdg  14557