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Theorem setrec2mpt 47930
Description: Version of setrec2 47928 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 5246 . 2 𝑎(𝑎𝐴𝑆)
2 setrec2mpt.1 . 2 𝐵 = setrecs((𝑎𝐴𝑆))
3 setrec2mpt.3 . . 3 (𝜑 → ∀𝑎(𝑎𝐶𝑆𝐶))
4 setrec2mpt.2 . . . . . . 7 (𝑎𝐴𝑆𝑉)
5 eqid 2724 . . . . . . . . 9 (𝑎𝐴𝑆) = (𝑎𝐴𝑆)
65fvmpt2 6999 . . . . . . . 8 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) = 𝑆)
7 eqimss 4032 . . . . . . . 8 (((𝑎𝐴𝑆)‘𝑎) = 𝑆 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
86, 7syl 17 . . . . . . 7 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
94, 8mpdan 684 . . . . . 6 (𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
105fvmptndm 7018 . . . . . . 7 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) = ∅)
11 0ss 4388 . . . . . . 7 ∅ ⊆ 𝑆
1210, 11eqsstrdi 4028 . . . . . 6 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
139, 12pm2.61i 182 . . . . 5 ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆
14 sstr2 3981 . . . . 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 1817 . 2 (𝜑 → ∀𝑎(𝑎𝐶 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝐶))
181, 2, 17setrec2 47928 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1531   = wceq 1533  wcel 2098  wss 3940  c0 4314  cmpt 5221  cfv 6533  setrecscsetrecs 47916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fv 6541  df-setrecs 47917
This theorem is referenced by: (None)
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