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Theorem sbcfg 6081
 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 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
Distinct variable groups:   𝑥,𝑉   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem sbcfg
StepHypRef Expression
1 df-f 5930 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
21a1i 11 . . 3 (𝑋𝑉 → (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
32sbcbidv 3523 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵[𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
4 sbcfng 6080 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
5 sbcssg 4118 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵))
6 csbrn 5631 . . . . . 6 𝑋 / 𝑥ran 𝐹 = ran 𝑋 / 𝑥𝐹
76sseq1i 3662 . . . . 5 (𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵 ↔ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵)
85, 7syl6bb 276 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵 ↔ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵))
94, 8anbi12d 747 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵) ↔ (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ∧ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵)))
10 sbcan 3511 . . 3 ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵))
11 df-f 5930 . . 3 (𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵 ↔ (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ∧ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵))
129, 10, 113bitr4g 303 . 2 (𝑋𝑉 → ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ 𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
133, 12bitrd 268 1 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2030  [wsbc 3468  ⦋csb 3566   ⊆ wss 3607  ran crn 5144   Fn wfn 5921  ⟶wf 5922 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928  df-fn 5929  df-f 5930 This theorem is referenced by:  csbwrdg  13366
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