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Mirrors > Home > MPE Home > Th. List > ovelrn | Structured version Visualization version GIF version |
Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.) |
Ref | Expression |
---|---|
ovelrn | ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrnov 7301 | . . 3 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}) | |
2 | 1 | eleq2d 2875 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)})) |
3 | ovex 7168 | . . . . . 6 ⊢ (𝑥𝐹𝑦) ∈ V | |
4 | eleq1 2877 | . . . . . 6 ⊢ (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V)) | |
5 | 3, 4 | mpbiri 261 | . . . . 5 ⊢ (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
6 | 5 | rexlimivw 3241 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
7 | 6 | rexlimivw 3241 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
8 | eqeq1 2802 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦))) | |
9 | 8 | 2rexbidv 3259 | . . 3 ⊢ (𝑧 = 𝐶 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦))) |
10 | 7, 9 | elab3 3622 | . 2 ⊢ (𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)) |
11 | 2, 10 | syl6bb 290 | 1 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 Vcvv 3441 × cxp 5517 ran crn 5520 Fn wfn 6319 (class class class)co 7135 |
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-sbc 3721 df-csb 3829 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-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 df-ov 7138 |
This theorem is referenced by: efgredlem 18865 efgcpbllemb 18873 gsumval3 19020 lecldbas 21824 blrnps 23015 blrn 23016 qdensere 23375 tgioo 23401 xrge0tsms 23439 ioorf 24177 ioorinv 24180 ioorcl 24181 dyaddisj 24200 dyadmax 24202 mbfid 24239 ismbfd 24243 hhssnv 29047 xrge0tsmsd 30742 iccllysconn 32610 rellysconn 32611 icoreelrnab 34771 relowlssretop 34780 relowlpssretop 34781 islptre 42261 |
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