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Mirrors > Home > MPE Home > Th. List > Mathboxes > finxpeq2 | Structured version Visualization version Unicode version |
Description: Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.) |
Ref | Expression |
---|---|
finxpeq2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2689 | . . . 4 | |
2 | opeq1 4402 | . . . . . . 7 | |
3 | rdgeq2 7508 | . . . . . . 7 | |
4 | 2, 3 | syl 17 | . . . . . 6 |
5 | id 22 | . . . . . 6 | |
6 | 4, 5 | fveq12d 6197 | . . . . 5 |
7 | 6 | eqeq2d 2632 | . . . 4 |
8 | 1, 7 | anbi12d 747 | . . 3 |
9 | 8 | abbidv 2741 | . 2 |
10 | df-finxp 33221 | . 2 | |
11 | df-finxp 33221 | . 2 | |
12 | 9, 10, 11 | 3eqtr4g 2681 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cab 2608 cvv 3200 c0 3915 cif 4086 cop 4183 cuni 4436 cxp 5112 cfv 5888 cmpt2 6652 com 7065 c1st 7166 crdg 7505 c1o 7553 cfinxp 33220 |
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 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 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-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fv 5896 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-finxp 33221 |
This theorem is referenced by: finxp2o 33236 finxp3o 33237 finxp00 33239 |
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