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Mirrors > Home > MPE Home > Th. List > elrnmpo | Structured version Visualization version GIF version |
Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rngop.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
elrnmpo.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elrnmpo | ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngop.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | rnmpo 7284 | . . 3 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} |
3 | 2 | eleq2i 2904 | . 2 ⊢ (𝐷 ∈ ran 𝐹 ↔ 𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}) |
4 | elrnmpo.1 | . . . . . 6 ⊢ 𝐶 ∈ V | |
5 | eleq1 2900 | . . . . . 6 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V)) | |
6 | 4, 5 | mpbiri 260 | . . . . 5 ⊢ (𝐷 = 𝐶 → 𝐷 ∈ V) |
7 | 6 | rexlimivw 3282 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V) |
8 | 7 | rexlimivw 3282 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V) |
9 | eqeq1 2825 | . . . 4 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶)) | |
10 | 9 | 2rexbidv 3300 | . . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)) |
11 | 8, 10 | elab3 3674 | . 2 ⊢ (𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶) |
12 | 3, 11 | bitri 277 | 1 ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 Vcvv 3494 ran crn 5556 ∈ cmpo 7158 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: qexALT 12364 lsmelvalx 18765 efgtlen 18852 frgpnabllem1 18993 fmucndlem 22900 mbfimaopnlem 24256 tglnunirn 26334 tpr2rico 31155 mbfmco2 31523 br2base 31527 dya2icobrsiga 31534 dya2iocnrect 31539 dya2iocucvr 31542 sxbrsigalem2 31544 cntotbnd 35089 eldiophb 39403 elicores 41858 volicorescl 42884 |
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