Theorem pimconstlt1 46359

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 4073	. . 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 483	. . . . . . 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 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
9	6, 8	fvmpt2d 7014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
10		pimconstlt1.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 < 𝐶)
11	10	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 < 𝐶)
12	9, 11	eqbrtrd 5167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) < 𝐶)
13	4, 12	jca 510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) < 𝐶))
14		rabid 3440	. . . . . 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 411	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}))
17	3, 16	ralrimi 3245	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶})
18		nfcv 2892	. . . 4 ⊢ Ⅎ𝑥𝐴
19		nfrab1 3439	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶}
20	18, 19	dfss3f 3970	. . 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 3996	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534  Ⅎwnf 1778   ∈ wcel 2099  ∀wral 3051  {crab 3419   ⊆ wss 3946   class class class wbr 5145   ↦ cmpt 5228  ‘cfv 6546  ℝcr 11148   < clt 11289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fv 6554

This theorem is referenced by:  smfconst  46406