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Mirrors > Home > MPE Home > Th. List > rnsnopg | Structured version Visualization version GIF version |
Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
rnsnopg | ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5275 | . . 3 ⊢ ran {〈𝐴, 𝐵〉} = dom ◡{〈𝐴, 𝐵〉} | |
2 | dfdm4 5469 | . . . 4 ⊢ dom {〈𝐵, 𝐴〉} = ran ◡{〈𝐵, 𝐴〉} | |
3 | df-rn 5275 | . . . 4 ⊢ ran ◡{〈𝐵, 𝐴〉} = dom ◡◡{〈𝐵, 𝐴〉} | |
4 | cnvcnvsn 5769 | . . . . 5 ⊢ ◡◡{〈𝐵, 𝐴〉} = ◡{〈𝐴, 𝐵〉} | |
5 | 4 | dmeqi 5478 | . . . 4 ⊢ dom ◡◡{〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} |
6 | 2, 3, 5 | 3eqtri 2784 | . . 3 ⊢ dom {〈𝐵, 𝐴〉} = dom ◡{〈𝐴, 𝐵〉} |
7 | 1, 6 | eqtr4i 2783 | . 2 ⊢ ran {〈𝐴, 𝐵〉} = dom {〈𝐵, 𝐴〉} |
8 | dmsnopg 5763 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom {〈𝐵, 𝐴〉} = {𝐵}) | |
9 | 7, 8 | syl5eq 2804 | 1 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐵〉} = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1630 ∈ wcel 2137 {csn 4319 〈cop 4325 ◡ccnv 5263 dom cdm 5264 ran crn 5265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pr 5053 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-rab 3057 df-v 3340 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-sn 4320 df-pr 4322 df-op 4326 df-br 4803 df-opab 4863 df-xp 5270 df-rel 5271 df-cnv 5272 df-dm 5274 df-rn 5275 |
This theorem is referenced by: rnpropg 5772 rnsnop 5774 funcnvpr 6109 funcnvtp 6110 dprdsn 18633 usgr1e 26334 1loopgredg 26605 1egrvtxdg0 26615 uspgrloopedg 26622 noextend 32123 rnsnf 39867 |
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