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Theorem sbcfg 5333
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 5189 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
21a1i 9 . . 3 (𝑋𝑉 → (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
32sbcbidv 3007 . 2 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵[𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)))
4 sbcfng 5332 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴))
5 sbcssg 3516 . . . . 5 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵))
6 csbrng 5062 . . . . . 6 (𝑋𝑉𝑋 / 𝑥ran 𝐹 = ran 𝑋 / 𝑥𝐹)
76sseq1d 3169 . . . . 5 (𝑋𝑉 → (𝑋 / 𝑥ran 𝐹𝑋 / 𝑥𝐵 ↔ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵))
85, 7bitrd 187 . . . 4 (𝑋𝑉 → ([𝑋 / 𝑥]ran 𝐹𝐵 ↔ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵))
94, 8anbi12d 465 . . 3 (𝑋𝑉 → (([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵) ↔ (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ∧ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵)))
10 sbcan 2991 . . 3 ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ ([𝑋 / 𝑥]𝐹 Fn 𝐴[𝑋 / 𝑥]ran 𝐹𝐵))
11 df-f 5189 . . 3 (𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵 ↔ (𝑋 / 𝑥𝐹 Fn 𝑋 / 𝑥𝐴 ∧ ran 𝑋 / 𝑥𝐹𝑋 / 𝑥𝐵))
129, 10, 113bitr4g 222 . 2 (𝑋𝑉 → ([𝑋 / 𝑥](𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ 𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
133, 12bitrd 187 1 (𝑋𝑉 → ([𝑋 / 𝑥]𝐹:𝐴𝐵𝑋 / 𝑥𝐹:𝑋 / 𝑥𝐴𝑋 / 𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2135  [wsbc 2949  csb 3043  wss 3114  ran crn 4602   Fn wfn 5180  wf 5181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2726  df-sbc 2950  df-csb 3044  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-br 3980  df-opab 4041  df-id 4268  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-fun 5187  df-fn 5188  df-f 5189
This theorem is referenced by:  ctiunctlemf  12365
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