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Theorem sbcfg 6734
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 6565 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
21a1i 11 . . 3 (𝑋𝑉 → (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
32sbcbidv 3845 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵[𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
4 sbcfng 6733 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
5 sbcssg 4520 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵))
6 csbrn 6223 . . . . . 6 𝑋 / 𝑥ran 𝐹 = ran 𝑋 / 𝑥𝐹
76sseq1i 4012 . . . . 5 (𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵 ↔ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵)
85, 7bitrdi 287 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵 ↔ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵))
94, 8anbi12d 632 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵) ↔ (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ∧ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵)))
10 sbcan 3838 . . 3 ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵))
11 df-f 6565 . . 3 (𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵 ↔ (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ∧ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵))
129, 10, 113bitr4g 314 . 2 (𝑋𝑉 → ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ 𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
133, 12bitrd 279 1 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  [wsbc 3788  csb 3899  wss 3951  ran crn 5686   Fn wfn 6556  wf 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  csbwrdg  14582
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