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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 7263 | . . 3 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} |
3 | 2 | eleq2i 2881 | . 2 ⊢ (𝐷 ∈ ran 𝐹 ↔ 𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶}) |
4 | elrnmpo.1 | . . . . . 6 ⊢ 𝐶 ∈ V | |
5 | eleq1 2877 | . . . . . 6 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V)) | |
6 | 4, 5 | mpbiri 261 | . . . . 5 ⊢ (𝐷 = 𝐶 → 𝐷 ∈ V) |
7 | 6 | rexlimivw 3241 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V) |
8 | 7 | rexlimivw 3241 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶 → 𝐷 ∈ V) |
9 | eqeq1 2802 | . . . 4 ⊢ (𝑧 = 𝐷 → (𝑧 = 𝐶 ↔ 𝐷 = 𝐶)) | |
10 | 9 | 2rexbidv 3259 | . . 3 ⊢ (𝑧 = 𝐷 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)) |
11 | 8, 10 | elab3 3622 | . 2 ⊢ (𝐷 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶) |
12 | 3, 11 | bitri 278 | 1 ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 Vcvv 3441 ran crn 5520 ∈ cmpo 7137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: qexALT 12351 lsmelvalx 18757 efgtlen 18844 frgpnabllem1 18986 fmucndlem 22897 mbfimaopnlem 24259 tglnunirn 26342 tpr2rico 31265 mbfmco2 31633 br2base 31637 dya2icobrsiga 31644 dya2iocnrect 31649 dya2iocucvr 31652 sxbrsigalem2 31654 cntotbnd 35234 eldiophb 39698 elicores 42170 volicorescl 43192 |
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