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Mirrors > Home > MPE Home > Th. List > ex-rn | Structured version Visualization version GIF version |
Description: Example for df-rn 5683. 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 5932 | . 2 ⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = ran {⟨2, 6⟩, ⟨3, 9⟩}) | |
2 | df-pr 4627 | . . . 4 ⊢ {⟨2, 6⟩, ⟨3, 9⟩} = ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) | |
3 | 2 | rneqi 5933 | . . 3 ⊢ ran {⟨2, 6⟩, ⟨3, 9⟩} = ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) |
4 | rnun 6144 | . . 3 ⊢ ran ({⟨2, 6⟩} ∪ {⟨3, 9⟩}) = (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) | |
5 | 2nn 12309 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
6 | 5 | elexi 3490 | . . . . . 6 ⊢ 2 ∈ V |
7 | 6 | rnsnop 6222 | . . . . 5 ⊢ ran {⟨2, 6⟩} = {6} |
8 | 3nn 12315 | . . . . . . 7 ⊢ 3 ∈ ℕ | |
9 | 8 | elexi 3490 | . . . . . 6 ⊢ 3 ∈ V |
10 | 9 | rnsnop 6222 | . . . . 5 ⊢ ran {⟨3, 9⟩} = {9} |
11 | 7, 10 | uneq12i 4157 | . . . 4 ⊢ (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = ({6} ∪ {9}) |
12 | df-pr 4627 | . . . 4 ⊢ {6, 9} = ({6} ∪ {9}) | |
13 | 11, 12 | eqtr4i 2759 | . . 3 ⊢ (ran {⟨2, 6⟩} ∪ ran {⟨3, 9⟩}) = {6, 9} |
14 | 3, 4, 13 | 3eqtri 2760 | . 2 ⊢ ran {⟨2, 6⟩, ⟨3, 9⟩} = {6, 9} |
15 | 1, 14 | eqtrdi 2784 | 1 ⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∪ cun 3943 {csn 4624 {cpr 4626 ⟨cop 4630 ran crn 5673 ℕcn 12236 2c2 12291 3c3 12292 6c6 12295 9c9 12298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 ax-1cn 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-nn 12237 df-2 12299 df-3 12300 |
This theorem is referenced by: (None) |
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