Theorem pimconstlt1 47130

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 4020	. . 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 484	. . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
9	6, 8	fvmpt2d 6961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
10		pimconstlt1.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 < 𝐶)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 < 𝐶)
12	9, 11	eqbrtrd 5107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) < 𝐶)
13	4, 12	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶))
14		rabid 3410	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶))
15	13, 14	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
16	15	ex 412	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}))
17	3, 16	ralrimi 3235	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
18		nfcv 2898	. . . 4 ⊢ Ⅎ𝑥𝐴
19		nfrab1 3409	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}
20	18, 19	dfss3f 3913	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
21	17, 20	sylibr 234	. 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
22	2, 21	eqssd 3939	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3051  {crab 3389   ⊆ wss 3889   class class class wbr 5085   ↦ cmpt 5166  ‘cfv 6498  ℝcr 11037   < clt 11179

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506

This theorem is referenced by:  smfconst  47177