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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 5648 | . . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 6113 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 5864 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | 1, 3 | eqtri 2761 | . 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
5 | dmxp 5888 | . 2 ⊢ (𝐴 ≠ ∅ → dom (𝐵 × 𝐴) = 𝐵) | |
6 | 4, 5 | eqtrid 2785 | 1 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ≠ wne 2940 ∅c0 4286 × cxp 5635 ◡ccnv 5636 dom cdm 5637 ran crn 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 |
This theorem is referenced by: rnxpid 6129 ssxpb 6130 xpima 6138 unixp 6238 fconst5 7159 rnmptc 7160 xpexr 7859 xpexr2 7860 fparlem3 8050 fparlem4 8051 frxp 8062 fodomr 9078 djuexb 9853 dfac5lem3 10069 fpwwe2lem12 10586 vdwlem8 16868 ramz 16905 gsumxp 19761 xkoccn 22993 txindislem 23007 cnextf 23440 metustexhalf 23935 ovolctb 24877 axlowdimlem13 27952 axlowdim1 27957 imadifxp 31572 sibf0 32998 ovoliunnfl 36170 voliunnfl 36172 |
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