Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mbfconstlem | Structured version Visualization version GIF version |
Description: Lemma for mbfconst 24163 and related theorems. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
mbfconstlem | ⊢ ((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvimass 5943 | . . . . . 6 ⊢ (◡(𝐴 × {𝐶}) “ 𝐵) ⊆ dom (𝐴 × {𝐶}) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → (◡(𝐴 × {𝐶}) “ 𝐵) ⊆ dom (𝐴 × {𝐶})) |
3 | cnvimarndm 5944 | . . . . . 6 ⊢ (◡(𝐴 × {𝐶}) “ ran (𝐴 × {𝐶})) = dom (𝐴 × {𝐶}) | |
4 | fconst6g 6562 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝐵 → (𝐴 × {𝐶}):𝐴⟶𝐵) | |
5 | 4 | adantl 482 | . . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → (𝐴 × {𝐶}):𝐴⟶𝐵) |
6 | frn 6514 | . . . . . . 7 ⊢ ((𝐴 × {𝐶}):𝐴⟶𝐵 → ran (𝐴 × {𝐶}) ⊆ 𝐵) | |
7 | imass2 5959 | . . . . . . 7 ⊢ (ran (𝐴 × {𝐶}) ⊆ 𝐵 → (◡(𝐴 × {𝐶}) “ ran (𝐴 × {𝐶})) ⊆ (◡(𝐴 × {𝐶}) “ 𝐵)) | |
8 | 5, 6, 7 | 3syl 18 | . . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → (◡(𝐴 × {𝐶}) “ ran (𝐴 × {𝐶})) ⊆ (◡(𝐴 × {𝐶}) “ 𝐵)) |
9 | 3, 8 | eqsstrrid 4015 | . . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → dom (𝐴 × {𝐶}) ⊆ (◡(𝐴 × {𝐶}) “ 𝐵)) |
10 | 2, 9 | eqssd 3983 | . . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → (◡(𝐴 × {𝐶}) “ 𝐵) = dom (𝐴 × {𝐶})) |
11 | fconstg 6560 | . . . . . 6 ⊢ (𝐶 ∈ ℝ → (𝐴 × {𝐶}):𝐴⟶{𝐶}) | |
12 | 11 | ad2antlr 723 | . . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → (𝐴 × {𝐶}):𝐴⟶{𝐶}) |
13 | 12 | fdmd 6517 | . . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → dom (𝐴 × {𝐶}) = 𝐴) |
14 | 10, 13 | eqtrd 2856 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → (◡(𝐴 × {𝐶}) “ 𝐵) = 𝐴) |
15 | simpll 763 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → 𝐴 ∈ dom vol) | |
16 | 14, 15 | eqeltrd 2913 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ∈ 𝐵) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol) |
17 | 11 | ad2antlr 723 | . . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ ¬ 𝐶 ∈ 𝐵) → (𝐴 × {𝐶}):𝐴⟶{𝐶}) |
18 | incom 4177 | . . . . 5 ⊢ ({𝐶} ∩ 𝐵) = (𝐵 ∩ {𝐶}) | |
19 | simpr 485 | . . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐵) | |
20 | disjsn 4641 | . . . . . 6 ⊢ ((𝐵 ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ 𝐵) | |
21 | 19, 20 | sylibr 235 | . . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ ¬ 𝐶 ∈ 𝐵) → (𝐵 ∩ {𝐶}) = ∅) |
22 | 18, 21 | syl5eq 2868 | . . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ ¬ 𝐶 ∈ 𝐵) → ({𝐶} ∩ 𝐵) = ∅) |
23 | fimacnvdisj 6551 | . . . 4 ⊢ (((𝐴 × {𝐶}):𝐴⟶{𝐶} ∧ ({𝐶} ∩ 𝐵) = ∅) → (◡(𝐴 × {𝐶}) “ 𝐵) = ∅) | |
24 | 17, 22, 23 | syl2anc 584 | . . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ ¬ 𝐶 ∈ 𝐵) → (◡(𝐴 × {𝐶}) “ 𝐵) = ∅) |
25 | 0mbl 24069 | . . 3 ⊢ ∅ ∈ dom vol | |
26 | 24, 25 | syl6eqel 2921 | . 2 ⊢ (((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) ∧ ¬ 𝐶 ∈ 𝐵) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol) |
27 | 16, 26 | pm2.61dan 809 | 1 ⊢ ((𝐴 ∈ dom vol ∧ 𝐶 ∈ ℝ) → (◡(𝐴 × {𝐶}) “ 𝐵) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4559 × cxp 5547 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 ⟶wf 6345 ℝcr 10525 volcvol 23993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-rp 12380 df-xadd 12498 df-ioo 12732 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-sum 15033 df-xmet 20468 df-met 20469 df-ovol 23994 df-vol 23995 |
This theorem is referenced by: ismbf 24158 mbfconst 24163 |
Copyright terms: Public domain | W3C validator |