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Theorem setrec2mpt 47644
Description: Version of setrec2 47642 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 5255 . 2 𝑎(𝑎𝐴𝑆)
2 setrec2mpt.1 . 2 𝐵 = setrecs((𝑎𝐴𝑆))
3 setrec2mpt.3 . . 3 (𝜑 → ∀𝑎(𝑎𝐶𝑆𝐶))
4 setrec2mpt.2 . . . . . . 7 (𝑎𝐴𝑆𝑉)
5 eqid 2733 . . . . . . . . 9 (𝑎𝐴𝑆) = (𝑎𝐴𝑆)
65fvmpt2 7005 . . . . . . . 8 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) = 𝑆)
7 eqimss 4039 . . . . . . . 8 (((𝑎𝐴𝑆)‘𝑎) = 𝑆 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
86, 7syl 17 . . . . . . 7 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
94, 8mpdan 686 . . . . . 6 (𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
105fvmptndm 7024 . . . . . . 7 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) = ∅)
11 0ss 4395 . . . . . . 7 ∅ ⊆ 𝑆
1210, 11eqsstrdi 4035 . . . . . 6 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
139, 12pm2.61i 182 . . . . 5 ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆
14 sstr2 3988 . . . . 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 1826 . 2 (𝜑 → ∀𝑎(𝑎𝐶 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝐶))
181, 2, 17setrec2 47642 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  wss 3947  c0 4321  cmpt 5230  cfv 6540  setrecscsetrecs 47630
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 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-setrecs 47631
This theorem is referenced by: (None)
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