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Theorem sbcfg 6204
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 6053 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
21a1i 11 . . 3 (𝑋𝑉 → (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
32sbcbidv 3631 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵[𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
4 sbcfng 6203 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
5 sbcssg 4229 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵))
6 csbrn 5754 . . . . . 6 𝑋 / 𝑥ran 𝐹 = ran 𝑋 / 𝑥𝐹
76sseq1i 3770 . . . . 5 (𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵 ↔ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵)
85, 7syl6bb 276 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵 ↔ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵))
94, 8anbi12d 749 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵) ↔ (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ∧ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵)))
10 sbcan 3619 . . 3 ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵))
11 df-f 6053 . . 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 2139  [wsbc 3576  csb 3674  wss 3715  ran crn 5267   Fn wfn 6044  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-fun 6051  df-fn 6052  df-f 6053
This theorem is referenced by:  csbwrdg  13520
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