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Theorem fnssintima 7382
Description: Condition for subset of an intersection of an image. (Contributed by Scott Fenton, 16-Aug-2024.)
Assertion
Ref Expression
fnssintima ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 (𝐹𝐵) ↔ ∀𝑥𝐵 𝐶 ⊆ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fnssintima
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssint 4964 . . 3 (𝐶 (𝐹𝐵) ↔ ∀𝑦 ∈ (𝐹𝐵)𝐶𝑦)
2 df-ral 3062 . . 3 (∀𝑦 ∈ (𝐹𝐵)𝐶𝑦 ↔ ∀𝑦(𝑦 ∈ (𝐹𝐵) → 𝐶𝑦))
31, 2bitri 275 . 2 (𝐶 (𝐹𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐹𝐵) → 𝐶𝑦))
4 fvelimab 6981 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
54imbi1d 341 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑦 ∈ (𝐹𝐵) → 𝐶𝑦) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦)))
65albidv 1920 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑦(𝑦 ∈ (𝐹𝐵) → 𝐶𝑦) ↔ ∀𝑦(∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦)))
7 ralcom4 3286 . . . 4 (∀𝑥𝐵𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑦𝑥𝐵 ((𝐹𝑥) = 𝑦𝐶𝑦))
8 eqcom 2744 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
98imbi1i 349 . . . . . . 7 (((𝐹𝑥) = 𝑦𝐶𝑦) ↔ (𝑦 = (𝐹𝑥) → 𝐶𝑦))
109albii 1819 . . . . . 6 (∀𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑦(𝑦 = (𝐹𝑥) → 𝐶𝑦))
11 fvex 6919 . . . . . . 7 (𝐹𝑥) ∈ V
12 sseq2 4010 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝐶𝑦𝐶 ⊆ (𝐹𝑥)))
1311, 12ceqsalv 3521 . . . . . 6 (∀𝑦(𝑦 = (𝐹𝑥) → 𝐶𝑦) ↔ 𝐶 ⊆ (𝐹𝑥))
1410, 13bitri 275 . . . . 5 (∀𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ 𝐶 ⊆ (𝐹𝑥))
1514ralbii 3093 . . . 4 (∀𝑥𝐵𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑥𝐵 𝐶 ⊆ (𝐹𝑥))
16 r19.23v 3183 . . . . 5 (∀𝑥𝐵 ((𝐹𝑥) = 𝑦𝐶𝑦) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦))
1716albii 1819 . . . 4 (∀𝑦𝑥𝐵 ((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑦(∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦))
187, 15, 173bitr3ri 302 . . 3 (∀𝑦(∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑥𝐵 𝐶 ⊆ (𝐹𝑥))
196, 18bitrdi 287 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑦(𝑦 ∈ (𝐹𝐵) → 𝐶𝑦) ↔ ∀𝑥𝐵 𝐶 ⊆ (𝐹𝑥)))
203, 19bitrid 283 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 (𝐹𝐵) ↔ ∀𝑥𝐵 𝐶 ⊆ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951   cint 4946  cima 5688   Fn wfn 6556  cfv 6561
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-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  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-uni 4908  df-int 4947  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-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  bday1s  27876  madebdaylemlrcut  27937
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