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Theorem sbcfung 4925
 Description: Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcfung (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))

Proof of Theorem sbcfung
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcan 2805 . . 3 ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ ([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
2 sbcrel 4426 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝐹 ↔ Rel 𝐴 / 𝑥𝐹))
3 sbcal 2810 . . . . 5 ([𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤[𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
4 sbcal 2810 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦[𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
5 sbcal 2810 . . . . . . . . 9 ([𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧))
6 sbcimg 2804 . . . . . . . . . . 11 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧)))
7 sbcan 2805 . . . . . . . . . . . . 13 ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) ↔ ([𝐴 / 𝑥]𝑤𝐹𝑦[𝐴 / 𝑥]𝑤𝐹𝑧))
8 sbcbrg 3813 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦))
9 csbconstg 2864 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
10 csbconstg 2864 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
119, 10breq12d 3777 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝑤𝐴 / 𝑥𝐹𝑦))
128, 11bitrd 177 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑦))
13 sbcbrg 3813 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧))
14 csbconstg 2864 . . . . . . . . . . . . . . . 16 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
159, 14breq12d 3777 . . . . . . . . . . . . . . 15 (𝐴𝑉 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐹𝐴 / 𝑥𝑧𝑤𝐴 / 𝑥𝐹𝑧))
1613, 15bitrd 177 . . . . . . . . . . . . . 14 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐹𝑧𝑤𝐴 / 𝑥𝐹𝑧))
1712, 16anbi12d 442 . . . . . . . . . . . . 13 (𝐴𝑉 → (([𝐴 / 𝑥]𝑤𝐹𝑦[𝐴 / 𝑥]𝑤𝐹𝑧) ↔ (𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧)))
187, 17syl5bb 181 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) ↔ (𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧)))
19 sbcg 2827 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝑧𝑦 = 𝑧))
2018, 19imbi12d 223 . . . . . . . . . . 11 (𝐴𝑉 → (([𝐴 / 𝑥](𝑤𝐹𝑦𝑤𝐹𝑧) → [𝐴 / 𝑥]𝑦 = 𝑧) ↔ ((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
216, 20bitrd 177 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2221albidv 1705 . . . . . . . . 9 (𝐴𝑉 → (∀𝑧[𝐴 / 𝑥]((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
235, 22syl5bb 181 . . . . . . . 8 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2423albidv 1705 . . . . . . 7 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥]𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
254, 24syl5bb 181 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
2625albidv 1705 . . . . 5 (𝐴𝑉 → (∀𝑤[𝐴 / 𝑥]𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
273, 26syl5bb 181 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
282, 27anbi12d 442 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]Rel 𝐹[𝐴 / 𝑥]𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧))))
291, 28syl5bb 181 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧))))
30 dffun2 4912 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
3130sbcbii 2818 . 2 ([𝐴 / 𝑥]Fun 𝐹[𝐴 / 𝑥](Rel 𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐹𝑦𝑤𝐹𝑧) → 𝑦 = 𝑧)))
32 dffun2 4912 . 2 (Fun 𝐴 / 𝑥𝐹 ↔ (Rel 𝐴 / 𝑥𝐹 ∧ ∀𝑤𝑦𝑧((𝑤𝐴 / 𝑥𝐹𝑦𝑤𝐴 / 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
3329, 31, 323bitr4g 212 1 (𝐴𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun 𝐴 / 𝑥𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   ∈ wcel 1393  [wsbc 2764  ⦋csb 2852   class class class wbr 3764  Rel wrel 4350  Fun wfun 4896 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-rel 4352  df-cnv 4353  df-co 4354  df-fun 4904 This theorem is referenced by:  sbcfng  5044
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