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Theorem setrec2mpt 47454
Description: Version of setrec2 47452 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 2732 . . . . . . . . 9 (𝑎𝐴𝑆) = (𝑎𝐴𝑆)
65fvmpt2 6996 . . . . . . . 8 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) = 𝑆)
7 eqimss 4037 . . . . . . . 8 (((𝑎𝐴𝑆)‘𝑎) = 𝑆 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
86, 7syl 17 . . . . . . 7 ((𝑎𝐴𝑆𝑉) → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
94, 8mpdan 685 . . . . . 6 (𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
105fvmptndm 7015 . . . . . . 7 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) = ∅)
11 0ss 4393 . . . . . . 7 ∅ ⊆ 𝑆
1210, 11eqsstrdi 4033 . . . . . 6 𝑎𝐴 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆)
139, 12pm2.61i 182 . . . . 5 ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝑆
14 sstr2 3986 . . . . 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 1825 . 2 (𝜑 → ∀𝑎(𝑎𝐶 → ((𝑎𝐴𝑆)‘𝑎) ⊆ 𝐶))
181, 2, 17setrec2 47452 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1539   = wceq 1541  wcel 2106  wss 3945  c0 4319  cmpt 5225  cfv 6533  setrecscsetrecs 47440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fv 6541  df-setrecs 47441
This theorem is referenced by: (None)
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