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Theorem sbcfg 6735
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 6567 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
21a1i 11 . . 3 (𝑋𝑉 → (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
32sbcbidv 3851 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵[𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
4 sbcfng 6734 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
5 sbcssg 4526 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵))
6 csbrn 6225 . . . . . 6 𝑋 / 𝑥ran 𝐹 = ran 𝑋 / 𝑥𝐹
76sseq1i 4024 . . . . 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 3844 . . 3 ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵))
11 df-f 6567 . . 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 2106  [wsbc 3791  csb 3908  wss 3963  ran crn 5690   Fn wfn 6558  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  csbwrdg  14579
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