Users' Mathboxes Mathbox for Emmett Weisz < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  setrec2mpt Structured version   Visualization version   GIF version

Theorem setrec2mpt 48928
Description: Version of setrec2 48926 where 𝐹 is defined using maps-to notation. Deduction form is omitted in the second hypothesis for simplicity. In practice, nothing important is lost since we are only interested in one choice of 𝐴, 𝑆, and 𝑉 at a time. However, we are interested in what happens when 𝐶 varies, so deduction form is used in the third hypothesis. (Contributed by Emmett Weisz, 4-Jun-2024.)
Hypotheses
Ref Expression
setrec2mpt.1 𝐵 = setrecs((𝑎𝐴𝑆))
setrec2mpt.2 (𝑎𝐴𝑆𝑉)
setrec2mpt.3 (𝜑 → ∀𝑎(𝑎𝐶𝑆𝐶))
Assertion
Ref Expression
setrec2mpt (𝜑𝐵𝐶)
Distinct variable groups:   𝐴,𝑎   𝐶,𝑎
Allowed substitution hints:   𝜑(𝑎)   𝐵(𝑎)   𝑆(𝑎)   𝑉(𝑎)

Proof of Theorem setrec2mpt
StepHypRef Expression
1 nfmpt1 5256 . 2 𝑎(𝑎𝐴𝑆)
2 setrec2mpt.1 . 2 𝐵 = setrecs((𝑎𝐴𝑆))
3 setrec2mpt.3 . . 3 (𝜑 → ∀𝑎(𝑎𝐶𝑆𝐶))
4 setrec2mpt.2 . . . . . . 7 (𝑎𝐴𝑆𝑉)
5 eqid 2735 . . . . . . . . 9 (𝑎𝐴𝑆) = (𝑎𝐴𝑆)
65fvmpt2 7027 . . . . . . . 8 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) = 𝑆)
7 eqimss 4054 . . . . . . . 8 (((𝑎𝐴𝑆)‘𝑎) = 𝑆 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
86, 7syl 17 . . . . . . 7 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
94, 8mpdan 687 . . . . . 6 (𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
105fvmptndm 7047 . . . . . . 7 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) = ∅)
11 0ss 4406 . . . . . . 7 ∅ ⊆ 𝑆
1210, 11eqsstrdi 4050 . . . . . 6 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
139, 12pm2.61i 182 . . . . 5 ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆
14 sstr2 4002 . . . . 5 (((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆 → (𝑆𝐶 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝐶))
1513, 14ax-mp 5 . . . 4 (𝑆𝐶 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝐶)
1615imim2i 16 . . 3 ((𝑎𝐶𝑆𝐶) → (𝑎𝐶 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝐶))
173, 16sylg 1820 . 2 (𝜑 → ∀𝑎(𝑎𝐶 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝐶))
181, 2, 17setrec2 48926 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  wss 3963  c0 4339  cmpt 5231  cfv 6563  setrecscsetrecs 48914
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
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-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  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-fv 6571  df-setrecs 48915
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator