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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 5644 | . . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 6109 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 5860 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | 1, 3 | eqtri 2764 | . 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
5 | dmxp 5884 | . 2 ⊢ (𝐴 ≠ ∅ → dom (𝐵 × 𝐴) = 𝐵) | |
6 | 4, 5 | eqtrid 2788 | 1 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ≠ wne 2943 ∅c0 4282 × cxp 5631 ◡ccnv 5632 dom cdm 5633 ran crn 5634 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-xp 5639 df-rel 5640 df-cnv 5641 df-dm 5643 df-rn 5644 |
This theorem is referenced by: rnxpid 6125 ssxpb 6126 xpima 6134 unixp 6234 fconst5 7155 rnmptc 7156 xpexr 7855 xpexr2 7856 fparlem3 8046 fparlem4 8047 frxp 8058 fodomr 9072 djuexb 9845 dfac5lem3 10061 fpwwe2lem12 10578 vdwlem8 16860 ramz 16897 gsumxp 19753 xkoccn 22970 txindislem 22984 cnextf 23417 metustexhalf 23912 ovolctb 24854 axlowdimlem13 27903 axlowdim1 27908 imadifxp 31519 sibf0 32934 ovoliunnfl 36120 voliunnfl 36122 |
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