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Mirrors > Home > MPE Home > Th. List > rnmpo | Structured version Visualization version GIF version |
Description: The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.) |
Ref | Expression |
---|---|
rngop.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
rnmpo | ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngop.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | df-mpo 7414 | . . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2761 | . . 3 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
4 | 3 | rneqi 5937 | . 2 ⊢ ran 𝐹 = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
5 | rnoprab2 7513 | . 2 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} | |
6 | 4, 5 | eqtri 2761 | 1 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3071 ran crn 5678 {coprab 7410 ∈ cmpo 7411 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-cnv 5685 df-dm 5687 df-rn 5688 df-oprab 7413 df-mpo 7414 |
This theorem is referenced by: elrnmpog 7544 elrnmpo 7545 ralrnmpo 7547 mpoexw 8065 dffi3 9426 ixpiunwdom 9585 qnnen 16156 txuni2 23069 txbas 23071 xkobval 23090 xkoopn 23093 txrest 23135 ptrescn 23143 tx1stc 23154 xkoptsub 23158 xkopt 23159 xkococn 23164 ptcmplem4 23559 met2ndci 24031 i1fadd 25212 i1fmul 25213 scutf 27313 mulsproplem9 27580 ssltmul1 27602 ssltmul2 27603 precsexlem11 27663 rnmposs 31899 cnre2csqima 32891 qqhval2 32962 icoreresf 36233 ptrest 36487 eldiophb 41495 |
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