Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimgtmnf2 | Structured version Visualization version Unicode version |
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound , is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
pimgtmnf2.1 | |
pimgtmnf2.2 |
Ref | Expression |
---|---|
pimgtmnf2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3687 | . . 3 | |
2 | 1 | a1i 11 | . 2 |
3 | ssid 3624 | . . . . 5 | |
4 | 3 | a1i 11 | . . . 4 |
5 | pimgtmnf2.2 | . . . . . . . 8 | |
6 | 5 | ffvelrnda 6359 | . . . . . . 7 |
7 | 6 | mnfltd 11958 | . . . . . 6 |
8 | 7 | ralrimiva 2966 | . . . . 5 |
9 | nfcv 2764 | . . . . . . 7 | |
10 | nfcv 2764 | . . . . . . 7 | |
11 | pimgtmnf2.1 | . . . . . . . 8 | |
12 | nfcv 2764 | . . . . . . . 8 | |
13 | 11, 12 | nffv 6198 | . . . . . . 7 |
14 | 9, 10, 13 | nfbr 4699 | . . . . . 6 |
15 | nfv 1843 | . . . . . 6 | |
16 | fveq2 6191 | . . . . . . 7 | |
17 | 16 | breq2d 4665 | . . . . . 6 |
18 | 14, 15, 17 | cbvral 3167 | . . . . 5 |
19 | 8, 18 | sylib 208 | . . . 4 |
20 | 4, 19 | jca 554 | . . 3 |
21 | nfcv 2764 | . . . 4 | |
22 | 21, 21 | ssrabf 39298 | . . 3 |
23 | 20, 22 | sylibr 224 | . 2 |
24 | 2, 23 | eqssd 3620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wnfc 2751 wral 2912 crab 2916 wss 3574 class class class wbr 4653 wf 5884 cfv 5888 cr 9935 cmnf 10072 clt 10074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 |
This theorem is referenced by: pimgtmnf 40932 |
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