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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 7528 | . . 3 ⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}) | |
2 | 1 | eleq2d 2820 | . 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)})) |
3 | ovex 7391 | . . . . . 6 ⊢ (𝑥𝐹𝑦) ∈ V | |
4 | eleq1 2822 | . . . . . 6 ⊢ (𝐶 = (𝑥𝐹𝑦) → (𝐶 ∈ V ↔ (𝑥𝐹𝑦) ∈ V)) | |
5 | 3, 4 | mpbiri 258 | . . . . 5 ⊢ (𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
6 | 5 | rexlimivw 3145 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
7 | 6 | rexlimivw 3145 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦) → 𝐶 ∈ V) |
8 | eqeq1 2737 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 = (𝑥𝐹𝑦) ↔ 𝐶 = (𝑥𝐹𝑦))) | |
9 | 8 | 2rexbidv 3210 | . . 3 ⊢ (𝑧 = 𝐶 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦))) |
10 | 7, 9 | elab3 3639 | . 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 205 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3070 Vcvv 3444 × cxp 5632 ran crn 5635 Fn wfn 6492 (class class class)co 7358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-ov 7361 |
This theorem is referenced by: efgredlem 19534 efgcpbllemb 19542 gsumval3 19689 lecldbas 22586 blrnps 23777 blrn 23778 qdensere 24149 tgioo 24175 xrge0tsms 24213 ioorf 24953 ioorinv 24956 ioorcl 24957 dyaddisj 24976 dyadmax 24978 mbfid 25015 ismbfd 25019 hhssnv 30248 xrge0tsmsd 31948 iccllysconn 33901 rellysconn 33902 icoreelrnab 35871 relowlssretop 35880 relowlpssretop 35881 islptre 43946 |
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