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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 7566 | . . 3 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} |
| 3 | 2 | eleq2i 2833 | . 2 ⊢ (𝐷 ∈ ran 𝐹 ↔ 𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}) |
| 4 | elrnmpo.1 | . . . . . 6 ⊢ 𝐶 ∈ V | |
| 5 | eleq1 2829 | . . . . . 6 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V)) | |
| 6 | 4, 5 | mpbiri 258 | . . . . 5 ⊢ (𝐷 = 𝐶 → 𝐷 ∈ V) |
| 7 | 6 | rexlimivw 3151 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V) |
| 8 | 7 | rexlimivw 3151 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V) |
| 9 | eqeq1 2741 | . . . 4 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶)) | |
| 10 | 9 | 2rexbidv 3222 | . . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)) |
| 11 | 8, 10 | elab3 3686 | . 2 ⊢ (𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶) |
| 12 | 3, 11 | bitri 275 | 1 ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 ran crn 5686 ∈ cmpo 7433 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: qexALT 13006 lsmelvalx 19658 efgtlen 19744 frgpnabllem1 19891 fmucndlem 24300 mbfimaopnlem 25690 tglnunirn 28556 tpr2rico 33911 mbfmco2 34267 br2base 34271 dya2icobrsiga 34278 dya2iocnrect 34283 dya2iocucvr 34286 sxbrsigalem2 34288 cntotbnd 37803 eldiophb 42768 elicores 45546 volicorescl 46568 |
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