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Mirrors > Home > MPE Home > Th. List > rnpropg | Structured version Visualization version 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 4572 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
2 | 1 | rneqi 5809 | . 2 ⊢ ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) |
3 | rnsnopg 6080 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝐴, 𝐶〉} = {𝐶}) | |
4 | 3 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {〈𝐴, 𝐶〉} = {𝐶}) |
5 | rnsnopg 6080 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → ran {〈𝐵, 𝐷〉} = {𝐷}) | |
6 | 5 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {〈𝐵, 𝐷〉} = {𝐷}) |
7 | 4, 6 | uneq12d 4142 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) = ({𝐶} ∪ {𝐷})) |
8 | rnun 6006 | . . 3 ⊢ ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = (ran {〈𝐴, 𝐶〉} ∪ ran {〈𝐵, 𝐷〉}) | |
9 | df-pr 4572 | . . 3 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
10 | 7, 8, 9 | 3eqtr4g 2883 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) = {𝐶, 𝐷}) |
11 | 2, 10 | syl5eq 2870 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 {csn 4569 {cpr 4571 〈cop 4575 ran crn 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 |
This theorem is referenced by: funcnvtp 6419 funcnvqp 6420 umgr2v2eedg 27308 esumsnf 31325 poimirlem9 34903 sge0sn 42668 |
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