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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 4969 . . 3 (𝐶 (𝐹𝐵) ↔ ∀𝑦 ∈ (𝐹𝐵)𝐶𝑦)
2 df-ral 3060 . . 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 1918 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑦(𝑦 ∈ (𝐹𝐵) → 𝐶𝑦) ↔ ∀𝑦(∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦)))
7 ralcom4 3284 . . . 4 (∀𝑥𝐵𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑦𝑥𝐵 ((𝐹𝑥) = 𝑦𝐶𝑦))
8 eqcom 2742 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
98imbi1i 349 . . . . . . 7 (((𝐹𝑥) = 𝑦𝐶𝑦) ↔ (𝑦 = (𝐹𝑥) → 𝐶𝑦))
109albii 1816 . . . . . 6 (∀𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑦(𝑦 = (𝐹𝑥) → 𝐶𝑦))
11 fvex 6920 . . . . . . 7 (𝐹𝑥) ∈ V
12 sseq2 4022 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝐶𝑦𝐶 ⊆ (𝐹𝑥)))
1311, 12ceqsalv 3519 . . . . . 6 (∀𝑦(𝑦 = (𝐹𝑥) → 𝐶𝑦) ↔ 𝐶 ⊆ (𝐹𝑥))
1410, 13bitri 275 . . . . 5 (∀𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ 𝐶 ⊆ (𝐹𝑥))
1514ralbii 3091 . . . 4 (∀𝑥𝐵𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑥𝐵 𝐶 ⊆ (𝐹𝑥))
16 r19.23v 3181 . . . . 5 (∀𝑥𝐵 ((𝐹𝑥) = 𝑦𝐶𝑦) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦))
1716albii 1816 . . . 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 1535   = wceq 1537  wcel 2106  wral 3059  wrex 3068  wss 3963   cint 4951  cima 5692   Fn wfn 6558  cfv 6563
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-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  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-uni 4913  df-int 4952  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-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  bday1s  27891  madebdaylemlrcut  27952
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