Theorem pimconstlt1 45408

Description: Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound larger than the constant, is the whole domain. First part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimconstlt1.1	⊢ Ⅎ𝑥𝜑
pimconstlt1.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
pimconstlt1.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
pimconstlt1.4	⊢ (𝜑 → 𝐵 < 𝐶)

Assertion

Ref	Expression
pimconstlt1	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem pimconstlt1

Step	Hyp	Ref	Expression
1		ssrab2 4077	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ⊆ 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ⊆ 𝐴)
3		pimconstlt1.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
5		pimconstlt1.3	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
7		pimconstlt1.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
8	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
9	6, 8	fvmpt2d 7011	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
10		pimconstlt1.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 < 𝐶)
11	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 < 𝐶)
12	9, 11	eqbrtrd 5170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) < 𝐶)
13	4, 12	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶))
14		rabid 3452	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶))
15	13, 14	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
16	15	ex 413	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}))
17	3, 16	ralrimi 3254	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
18		nfcv 2903	. . . 4 ⊢ Ⅎ𝑥𝐴
19		nfrab1 3451	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}
20	18, 19	dfss3f 3973	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
21	17, 20	sylibr 233	. 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
22	2, 21	eqssd 3999	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  ∀wral 3061  {crab 3432   ⊆ wss 3948   class class class wbr 5148   ↦ cmpt 5231  ‘cfv 6543  ℝcr 11108   < clt 11247

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-sep 5299  ax-nul 5306  ax-pr 5427

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-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551

This theorem is referenced by:  smfconst  45455