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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 7605 | . . 3 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}) | |
2 | 1 | eleq2d 2824 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)})) |
3 | ovex 7463 | . . . . . 6 ⊢ (𝑥𝐹𝑦) ∈ V | |
4 | eleq1 2826 | . . . . . 6 ⊢ (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V)) | |
5 | 3, 4 | mpbiri 258 | . . . . 5 ⊢ (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
6 | 5 | rexlimivw 3148 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
7 | 6 | rexlimivw 3148 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
8 | eqeq1 2738 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦))) | |
9 | 8 | 2rexbidv 3219 | . . 3 ⊢ (𝑧 = 𝐶 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦))) |
10 | 7, 9 | elab3 3688 | . 2 ⊢ (𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦)) |
11 | 2, 10 | bitrdi 287 | 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 206 = wceq 1536 ∈ wcel 2105 {cab 2711 ∃wrex 3067 Vcvv 3477 × cxp 5686 ran crn 5689 Fn wfn 6557 (class class class)co 7430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 df-ov 7433 |
This theorem is referenced by: efgredlem 19779 efgcpbllemb 19787 gsumval3 19939 lecldbas 23242 blrnps 24433 blrn 24434 qdensere 24805 tgioo 24831 xrge0tsms 24869 ioorf 25621 ioorinv 25624 ioorcl 25625 dyaddisj 25644 dyadmax 25646 mbfid 25683 ismbfd 25687 hhssnv 31292 xrge0tsmsd 33047 iccllysconn 35234 rellysconn 35235 icoreelrnab 37336 relowlssretop 37345 relowlpssretop 37346 islptre 45574 |
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