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Mirrors > Home > MPE Home > Th. List > rnxp | Structured version Visualization version GIF version |
Description: The range of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
Ref | Expression |
---|---|
rnxp | ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5530 | . . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 5981 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 5737 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | 1, 3 | eqtri 2821 | . 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
5 | dmxp 5763 | . 2 ⊢ (𝐴 ≠ ∅ → dom (𝐵 × 𝐴) = 𝐵) | |
6 | 4, 5 | syl5eq 2845 | 1 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ≠ wne 2987 ∅c0 4243 × cxp 5517 ◡ccnv 5518 dom cdm 5519 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: rnxpid 5997 ssxpb 5998 xpima 6006 unixp 6101 fconst5 6945 rnmptc 6946 xpexr 7605 xpexr2 7606 fparlem3 7792 fparlem4 7793 frxp 7803 fodomr 8652 djuexb 9322 dfac5lem3 9536 fpwwe2lem13 10053 vdwlem8 16314 ramz 16351 gsumxp 19089 xkoccn 22224 txindislem 22238 cnextf 22671 metustexhalf 23163 ovolctb 24094 axlowdimlem13 26748 axlowdim1 26753 imadifxp 30364 sibf0 31702 ovoliunnfl 35099 voliunnfl 35101 |
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