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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptbd2 | Structured version Visualization version Unicode version |
Description: Boundness below of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
rnmptbd2.x | |
rnmptbd2.b |
Ref | Expression |
---|---|
rnmptbd2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4656 | . . . . 5 | |
2 | 1 | ralbidv 2986 | . . . 4 |
3 | 2 | cbvrexv 3172 | . . 3 |
4 | 3 | a1i 11 | . 2 |
5 | rnmptbd2.x | . . 3 | |
6 | rnmptbd2.b | . . 3 | |
7 | 5, 6 | rnmptbd2lem 39463 | . 2 |
8 | breq1 4656 | . . . . . 6 | |
9 | 8 | ralbidv 2986 | . . . . 5 |
10 | breq2 4657 | . . . . . . 7 | |
11 | 10 | cbvralv 3171 | . . . . . 6 |
12 | 11 | a1i 11 | . . . . 5 |
13 | 9, 12 | bitrd 268 | . . . 4 |
14 | 13 | cbvrexv 3172 | . . 3 |
15 | 14 | a1i 11 | . 2 |
16 | 4, 7, 15 | 3bitrd 294 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wnf 1708 wcel 1990 wral 2912 wrex 2913 class class class wbr 4653 cmpt 4729 crn 5115 cr 9935 cle 10075 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-mpt 4730 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: limsupvaluz2 39970 |
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