Theorem pimltpnf2 41440

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimltpnf2.1	⊢ Ⅎ𝑥𝐹
pimltpnf2.2	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)

Assertion

Ref	Expression
pimltpnf2	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem pimltpnf2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2913	. . . 4 ⊢ Ⅎ𝑥𝐴
2		nfcv 2913	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfv 1995	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞
4		pimltpnf2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2913	. . . . . 6 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6341	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2913	. . . . 5 ⊢ Ⅎ𝑥 <
8		nfcv 2913	. . . . 5 ⊢ Ⅎ𝑥+∞
9	6, 7, 8	nfbr 4834	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞
10		fveq2 6333	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq1d 4797	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞))
12	1, 2, 3, 9, 11	cbvrab 3348	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞}
13	12	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞})
14		nfv 1995	. . 3 ⊢ Ⅎ𝑦𝜑
15		pimltpnf2.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
16	15	ffvelrnda 6504	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
17	14, 16	pimltpnf 41433	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴)
18	13, 17	eqtrd 2805	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Syntax hints:   → wi 4   = wceq 1631  Ⅎwnfc 2900  {crab 3065   class class class wbr 4787  ⟶wf 6026  ‘cfv 6030  ℝcr 10141  +∞cpnf 10277   < clt 10280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-pnf 10282  df-xr 10284  df-ltxr 10285

This theorem is referenced by:  smfpimltxr  41473