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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 5688 | . . 3 ⊢ ran (𝐴 × 𝐵) = dom ◡(𝐴 × 𝐵) | |
2 | cnvxp 6157 | . . . 4 ⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴) | |
3 | 2 | dmeqi 5905 | . . 3 ⊢ dom ◡(𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
4 | 1, 3 | eqtri 2761 | . 2 ⊢ ran (𝐴 × 𝐵) = dom (𝐵 × 𝐴) |
5 | dmxp 5929 | . 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 2941 ∅c0 4323 × cxp 5675 ◡ccnv 5676 dom cdm 5677 ran crn 5678 |
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 5300 ax-nul 5307 ax-pr 5428 |
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 2942 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 df-rn 5688 |
This theorem is referenced by: rnxpid 6173 ssxpb 6174 xpima 6182 unixp 6282 fconst5 7207 rnmptc 7208 xpexr 7909 xpexr2 7910 fparlem3 8100 fparlem4 8101 frxp 8112 fodomr 9128 djuexb 9904 dfac5lem3 10120 fpwwe2lem12 10637 vdwlem8 16921 ramz 16958 gsumxp 19844 xkoccn 23123 txindislem 23137 cnextf 23570 metustexhalf 24065 ovolctb 25007 axlowdimlem13 28212 axlowdim1 28217 imadifxp 31832 sibf0 33333 ovoliunnfl 36530 voliunnfl 36532 |
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