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Mirrors > Home > ILE Home > Th. List > rnpropg | GIF version |
Description: The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Ref | Expression |
---|---|
rnpropg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3382 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
2 | 1 | rneqi 4562 | . 2 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) |
3 | rnsnopg 4799 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐶〉} = {𝐶}) | |
4 | 3 | adantr 261 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {〈𝐴, 𝐶〉} = {𝐶}) |
5 | rnsnopg 4799 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → ran {〈𝐵, 𝐷〉} = {𝐷}) | |
6 | 5 | adantl 262 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {〈𝐵, 𝐷〉} = {𝐷}) |
7 | 4, 6 | uneq12d 3098 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = ({𝐶} ∪ {𝐷})) |
8 | rnun 4732 | . . 3 ⊢ ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) | |
9 | df-pr 3382 | . . 3 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
10 | 7, 8, 9 | 3eqtr4g 2097 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = {𝐶, 𝐷}) |
11 | 2, 10 | syl5eq 2084 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∪ cun 2915 {csn 3375 {cpr 3376 〈cop 3378 ran crn 4346 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 |
This theorem is referenced by: (None) |
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