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Theorem setrec2mpt 49216
Description: Version of setrec2 49214 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 5250 . 2 𝑎(𝑎𝐴𝑆)
2 setrec2mpt.1 . 2 𝐵 = setrecs((𝑎𝐴𝑆))
3 setrec2mpt.3 . . 3 (𝜑 → ∀𝑎(𝑎𝐶𝑆𝐶))
4 setrec2mpt.2 . . . . . . 7 (𝑎𝐴𝑆𝑉)
5 eqid 2737 . . . . . . . . 9 (𝑎𝐴𝑆) = (𝑎𝐴𝑆)
65fvmpt2 7027 . . . . . . . 8 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) = 𝑆)
7 eqimss 4042 . . . . . . . 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 4400 . . . . . . 7 ∅ ⊆ 𝑆
1210, 11eqsstrdi 4028 . . . . . 6 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
139, 12pm2.61i 182 . . . . 5 ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆
14 sstr2 3990 . . . . 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 1823 . 2 (𝜑 → ∀𝑎(𝑎𝐶 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝐶))
181, 2, 17setrec2 49214 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1538   = wceq 1540  wcel 2108  wss 3951  c0 4333  cmpt 5225  cfv 6561  setrecscsetrecs 49202
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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  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-fv 6569  df-setrecs 49203
This theorem is referenced by: (None)
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