Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > preimaioomnf | Structured version Visualization version GIF version |
Description: Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
preimaioomnf.1 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
preimaioomnf.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
preimaioomnf | ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimaioomnf.1 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
2 | 1 | ffund 6521 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
3 | 1 | frnd 6524 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
4 | fimacnvinrn2 6844 | . . . 4 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ ℝ) → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) | |
5 | 2, 3, 4 | syl2anc 586 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ))) |
6 | preimaioomnf.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
7 | 6 | icomnfinre 41834 | . . . 4 ⊢ (𝜑 → ((-∞[,)𝐵) ∩ ℝ) = (-∞(,)𝐵)) |
8 | 7 | imaeq2d 5932 | . . 3 ⊢ (𝜑 → (◡𝐹 “ ((-∞[,)𝐵) ∩ ℝ)) = (◡𝐹 “ (-∞(,)𝐵))) |
9 | 5, 8 | eqtr2d 2860 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = (◡𝐹 “ (-∞[,)𝐵))) |
10 | 1 | frexr 41661 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
11 | 10, 6 | preimaicomnf 42997 | . 2 ⊢ (𝜑 → (◡𝐹 “ (-∞[,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
12 | 9, 11 | eqtrd 2859 | 1 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {crab 3145 ∩ cin 3938 ⊆ wss 3939 class class class wbr 5069 ◡ccnv 5557 ran crn 5559 “ cima 5561 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 -∞cmnf 10676 ℝ*cxr 10677 < clt 10678 (,)cioo 12741 [,)cico 12743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-pre-lttri 10614 ax-pre-lttrn 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-ioo 12745 df-ico 12747 |
This theorem is referenced by: issmflem 43011 mbfresmf 43023 smfres 43072 smfco 43084 |
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