Theorem setrec2mpt 49686

Description: Version of setrec2 49684 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 5206	. 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 2729	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 ↦ 𝑆) = (𝑎 ∈ 𝐴 ↦ 𝑆)
6	5	fvmpt2 6979	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = 𝑆)
7		eqimss 4005	. . . . . . . 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 6999	. . . . . . 7 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = ∅)
11		0ss 4363	. . . . . . 7 ⊢ ∅ ⊆ 𝑆
12	10, 11	eqsstrdi 3991	. . . . . 6 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
13	9, 12	pm2.61i 182	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆
14		sstr2 3953	. . . . 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 1823	. 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶))
18	1, 2, 17	setrec2 49684	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  ∅c0 4296   ↦ cmpt 5188  ‘cfv 6511  setrecscsetrecs 49672

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-setrecs 49673

This theorem is referenced by: (None)