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Mirrors > Home > MPE Home > Th. List > ex-rn | Structured version Visualization version GIF version |
Description: Example for df-rn 5649. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.) |
Ref | Expression |
---|---|
ex-rn | ⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rneq 5896 | . 2 ⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = ran {⟨2, 6⟩, ⟨3, 9⟩}) | |
2 | df-pr 4594 | . . . 4 ⊢ {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) | |
3 | 2 | rneqi 5897 | . . 3 ⊢ ran {⟨2, 6⟩, ⟨3, 9⟩} = ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) |
4 | rnun 6103 | . . 3 ⊢ ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) | |
5 | 2nn 12233 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
6 | 5 | elexi 3467 | . . . . . 6 ⊢ 2 ∈ V |
7 | 6 | rnsnop 6181 | . . . . 5 ⊢ ran {⟨2, 6⟩} = {6} |
8 | 3nn 12239 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
9 | 8 | elexi 3467 | . . . . . 6 ⊢ 3 ∈ V |
10 | 9 | rnsnop 6181 | . . . . 5 ⊢ ran {⟨3, 9⟩} = {9} |
11 | 7, 10 | uneq12i 4126 | . . . 4 ⊢ (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = ({6} ∪ {9}) |
12 | df-pr 4594 | . . . 4 ⊢ {6, 9} = ({6} ∪ {9}) | |
13 | 11, 12 | eqtr4i 2768 | . . 3 ⊢ (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = {6, 9} |
14 | 3, 4, 13 | 3eqtri 2769 | . 2 ⊢ ran {⟨2, 6⟩, ⟨3, 9⟩} = {6, 9} |
15 | 1, 14 | eqtrdi 2793 | 1 ⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∪ cun 3913 {csn 4591 {cpr 4593 ⟨cop 4597 ran crn 5639 ℕcn 12160 2c2 12215 3c3 12216 6c6 12219 9c9 12222 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 ax-1cn 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-nn 12161 df-2 12223 df-3 12224 |
This theorem is referenced by: (None) |
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