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Theorem fnssintima 7308
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 4926 . . 3 (𝐶 (𝐹𝐵) ↔ ∀𝑦 ∈ (𝐹𝐵)𝐶𝑦)
2 df-ral 3066 . . 3 (∀𝑦 ∈ (𝐹𝐵)𝐶𝑦 ↔ ∀𝑦(𝑦 ∈ (𝐹𝐵) → 𝐶𝑦))
31, 2bitri 275 . 2 (𝐶 (𝐹𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐹𝐵) → 𝐶𝑦))
4 fvelimab 6915 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
54imbi1d 342 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑦 ∈ (𝐹𝐵) → 𝐶𝑦) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦)))
65albidv 1924 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑦(𝑦 ∈ (𝐹𝐵) → 𝐶𝑦) ↔ ∀𝑦(∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦)))
7 ralcom4 3270 . . . 4 (∀𝑥𝐵𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑦𝑥𝐵 ((𝐹𝑥) = 𝑦𝐶𝑦))
8 eqcom 2744 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
98imbi1i 350 . . . . . . 7 (((𝐹𝑥) = 𝑦𝐶𝑦) ↔ (𝑦 = (𝐹𝑥) → 𝐶𝑦))
109albii 1822 . . . . . 6 (∀𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑦(𝑦 = (𝐹𝑥) → 𝐶𝑦))
11 fvex 6856 . . . . . . 7 (𝐹𝑥) ∈ V
12 sseq2 3971 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝐶𝑦𝐶 ⊆ (𝐹𝑥)))
1311, 12ceqsalv 3482 . . . . . 6 (∀𝑦(𝑦 = (𝐹𝑥) → 𝐶𝑦) ↔ 𝐶 ⊆ (𝐹𝑥))
1410, 13bitri 275 . . . . 5 (∀𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ 𝐶 ⊆ (𝐹𝑥))
1514ralbii 3097 . . . 4 (∀𝑥𝐵𝑦((𝐹𝑥) = 𝑦𝐶𝑦) ↔ ∀𝑥𝐵 𝐶 ⊆ (𝐹𝑥))
16 r19.23v 3180 . . . . 5 (∀𝑥𝐵 ((𝐹𝑥) = 𝑦𝐶𝑦) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦𝐶𝑦))
1716albii 1822 . . . 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 205  wa 397  wal 1540   = wceq 1542  wcel 2107  wral 3065  wrex 3074  wss 3911   cint 4908  cima 5637   Fn wfn 6492  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  bday1s  27173  madebdaylemlrcut  27231
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