Theorem setrec2mpt 50516

Description: Version of setrec2 50514 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 5215	. 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 2766	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐴 ↦ 𝑆) = (𝑎 ∈ 𝐴 ↦ 𝑆)
6	5	fvmpt2 7008	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = 𝑆)
7		eqimss 3998	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = 𝑆 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
8	6, 7	syl 18	. . . . . . 7 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
9	4, 8	mpdan 700	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
10	5	fvmptndm 7028	. . . . . . 7 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) = ∅)
11		0ss 4360	. . . . . . 7 ⊢ ∅ ⊆ 𝑆
12	10, 11	eqsstrdi 3984	. . . . . 6 ⊢ (¬ 𝑎 ∈ 𝐴 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆)
13	9, 12	pm2.61i 184	. . . . 5 ⊢ ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝑆
14		sstr2 3947	. . . . 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 17	. . 3 ⊢ ((𝑎 ⊆ 𝐶 → 𝑆 ⊆ 𝐶) → (𝑎 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶))
17	3, 16	sylg 1856	. 2 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐶 → ((𝑎 ∈ 𝐴 ↦ 𝑆)‘𝑎) ⊆ 𝐶))
18	1, 2, 17	setrec2 50514	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146   ⊆ wss 3908  ∅c0 4289   ↦ cmpt 5197  ‘cfv 6543  setrecscsetrecs 50502

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  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 6499  df-fun 6545  df-fv 6551  df-setrecs 50503

This theorem is used by: (None)