Theorem setrec2mpt 49812

Description: Version of setrec2 49810 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((𝑎 ∈ 𝐴 ↦ 𝑆))
setrec2mpt.2	⊢ (𝑎 ∈ 𝐴 → 𝑆 ∈ 𝑉)
setrec2mpt.3	⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → 𝑆 ⊆ 𝐶))

Assertion

Ref	Expression
setrec2mpt	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Distinct variable groups:   𝐴,𝑎   𝐶,𝑎

Allowed substitution hints:   𝜑(𝑎)   𝐵(𝑎)   𝑆(𝑎)   𝑉(𝑎)

Proof of Theorem setrec2mpt

Step	Hyp	Ref	Expression
1		nfmpt1 5194	. 2 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴 ↦ 𝑆)
2		setrec2mpt.1	. 2 ⊢ 𝐵 = setrecs((𝑎 ∈ 𝐴 ↦ 𝑆))
3		setrec2mpt.3	. . 3 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → 𝑆 ⊆ 𝐶))
4		setrec2mpt.2	. . . . . . 7 ⊢ (𝑎 ∈ 𝐴 → 𝑆 ∈ 𝑉)
5		eqid 2733	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 ↦ 𝑆) = (𝑎 ∈ 𝐴 ↦ 𝑆)
6	5	fvmpt2 6949	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = 𝑆)
7		eqimss 3990	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = 𝑆 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
9	4, 8	mpdan 687	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
10	5	fvmptndm 6969	. . . . . . 7 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = ∅)
11		0ss 4351	. . . . . . 7 ⊢ ∅ ⊆ 𝑆
12	10, 11	eqsstrdi 3976	. . . . . 6 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
13	9, 12	pm2.61i 182	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆
14		sstr2 3938	. . . . 5 ⊢ (((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆 → (𝑆 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶))
15	13, 14	ax-mp 5	. . . 4 ⊢ (𝑆 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶)
16	15	imim2i 16	. . 3 ⊢ ((𝑎 ⊆ 𝐶 → 𝑆 ⊆ 𝐶) → (𝑎 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶))
17	3, 16	sylg 1824	. 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶))
18	1, 2, 17	setrec2 49810	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113   ⊆ wss 3899  ∅c0 4284   ↦ cmpt 5176  ‘cfv 6489  setrecscsetrecs 49798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fv 6497  df-setrecs 49799

This theorem is referenced by: (None)